hyperbolic tessellations and k3 - king's college londonnumber conjecture that was formulated...

HYPERBOLIC TESSELLATIONS

AND GENERATORS OF K3

FOR IMAGINARY QUADRATIC FIELDS

DAVID BURNS, ROB DE JEU, HERBERT GANGL, ALEXANDER D. RAHM, AND DAN YASAKI

Abstract. We develop methods for constructing explicit generating elements, modulo torsion,of the K3-groups of imaginary quadratic number fields. These methods are based on eithertessellations of hyperbolic 3-space or on direct calculations in suitable pre-Bloch groups, andlead to the first proven examples of explicit generators, modulo torsion, of any infinite K3-groupof a number field. As part of this approach, we also make several improvements to the theoryof Bloch groups for K3 of any infinite field, predict the precise power of 2 that should occur inthe Lichtenbaum conjecture at −1 and verify that the latter prediction is valid for all abeliannumber fields.

Contents

1. Introduction 21.1. The general context 21.2. The main results 31.3. Notations and conventions 51.4. Acknowledgements 52. The conjectures of Lichtenbaum and of Bloch and Kato 52.1. Statement of the main result 62.2. The proof of Theorem 2.3: a first reduction 82.3. The role of Chern class maps 82.4. Pairings on Betti cohomology and Beilinson’s regulator 112.5. Completion of the proof of Theorem 2.3 122.6. An upper bound for t2m(F ) 143. K-theory, wedge complexes, and configurations of points 163.1. Towards explicit versions of K3(F) and reg2 173.2. Analysing the twisted wedge product. 183.3. Configurations of points, and a modified Bloch group. 213.4. Torsion elements in Bloch groups 263.5. A conjectural link between the groups of Bloch and Suslin 294. A geometric construction of elements in the modified Bloch group 304.1. Voronoi theory of Hermitian forms 304.2. Bloch elements from ideal tessellations of hyperbolic space 32

2010 Mathematics Subject Classification. Primary: 11R70, 19F27, 19D45; secondary: 11R42, 20H20, 51M20.Key words and phrases. regulator, Lichtenbaum conjecture, Bloch group, imaginary quadratic field, hyperbolic

3-space, tessellations, Bianchi group.

1

2 DAVID BURNS, ROB DE JEU, HERBERT GANGL, ALEXANDER D. RAHM, AND DAN YASAKI

4.3. Regulator maps and K-theory 334.4. A cyclotomic description of βgeo 345. The proof of Theorem 4.7 355.1. A preliminary result concerning orbits 355.2. The proof 365.3. An explicit example 416. Finding a generator of K3(k), and computing |K2(Ok)| 436.1. Dividing βgeo by |K2(Ok)| 43

6.2. Finding a generator of K3(k)indtf directly 45

6.3. Constructing elements in ker(δ2,k) via exceptional S-units 45Appendix A. Orders of finite subgroups 47Appendix B. A generator of K3(k)ind

tf for k = Q(√−303) 49

References 52

1. Introduction

1.1. The general context. Let F be a number field with ring of algebraic integers OF . Then,for each natural number m with m > 1, the algebraic K-group Km(OF ) in degree m of Quillenis a fundamental invariant of F , constituting a natural generalization in even degrees of the idealclass group of OF and in odd degrees of the group of units of OF .

Following fundamental work of Quillen [37], and of Borel [5], the abelian groups Km(OF ) areknown to be finite in positive even degrees and finitely generated in all degrees.

In addition, as a natural generalization of the analytic class number formula, Lichtenbaum [35]has conjectured that the leading coefficient ζ∗F (1−m) in the Taylor expansion at s = 1−m ofthe Riemann zeta function ζF (s) of F should satisfy

(1.1) ζ∗F (1−m) = ±2nm,F|K2m−2(OF )||K2m−1(OF )tor|

Rm(F ).

Here we write |X| for the cardinality of a finite set X, K2m−1(OF )tor for the torsion subgroup ofK2m−1(OF ), Rm(F ) for the covolume of the image of K2m−1(OF ) under the Beilinson regulatormap, and nm,F for an undetermined integer.

However, whilst Borel [6] has proved that the quotient of ζ∗F (1−m) by Rm(F ) is always rational(see Theorem 2.1), the only family of fields F for which (1.1) is known to be unconditionallyvalid are abelian extensions of Q (cf. Remark 2.10).

In addition, it has remained a difficult problem to explicitly compute, except in special cases,either |K2m−2(OF )| or Rm(F ) or to describe explicit generators of K2m−1(OF ) modulo torsion.

These are then the main problems that motivated the present article. To address them weare led to clarify the integer nm,F that should occur in (1.1), to investigate the fine integralproperties of various Bloch groups, to develop techniques for checking divisibility in such groupsand, in the case of imaginary quadratic fields F (which are the number fields of lowest degree forwhich the group K3(OF ) is infinite), to use ideal tessellations of hyperbolic 3-space to explicitlyconstruct elements in a suitable Bloch group that can be shown to generate a subgroup of index|K2(OF )| in the quotient group K3(OF )/K3(OF )tor.

HYPERBOLIC TESSELLATIONS AND K3 3

Whilst each of these aspects is perhaps of independent interest, it is only by understandingthe precise interplay between them that we are able to make significant progress on the aboveproblems.

In fact, as far as we are aware, our study is the first in which the links between these theorieshave been investigated in any precise, or systematic, way in the setting of integral (ratherthan rational) coefficients, and it seems likely that further analysis in this direction can lead toadditional insights.

At this stage, at least, we have used the approach developed here to explicitly determinefor all imaginary quadratic fields k of absolute discriminant at most 1000 a generator, modulotorsion, of K3(Ok), the order of K2(Ok) and the value of the Beilinson regulator R2(k).

In this way we have obtained the first proven examples of explicit generators, modulo torsion,of the K3-group of any number field for which the group is infinite, thereby resolving a problemthat has been considered ever since K3-groups were first introduced. We have also determinedthe order of K2(Ok) in a number of interesting new cases.

These computations both rely on, and complement, earlier work of Belabas and the thirdauthor in [2] concerning the orders of groups of the form K2(Ok). In particular, our computationsshow that the (divisional) bounds on |K2(Ok)| that are obtained in loc. cit. (in those cases wherethe order could not be precisely established) are sharp, and also prove an earlier associatedconjecture of Browkin and the third author from [10].

In a different direction, our methods have also given the first concrete evidence to suggestboth that the groups defined by Suslin (in terms of group homology) in [42] and by Bloch (interms of relative K-theory) in [4] should be related in a very natural way, and also that Bloch’sgroup should account for all of the indecomposable K3-group, at least modulo torsion.

In addition, some of the techniques developed here extend to number fields F that need notbe abelian (or even Galois) over Q, in which case essentially nothing of a general nature beyondthe result of Borel is known about Lichtenbaum’s conjectural formula (1.1).

For such fields our methods can in principle be used to either construct elements that generatea finite index subgroup of K3(OF ) and so can be used to investigate the possible validity of (1.1)or, if the more precise form of (1.1) discussed below is known to be valid for F and m = 2, toconstruct a full set of generators, modulo torsion, of K3(OF ).

For the sake of brevity, however, we shall not pursue these aspects in the present article.

1.2. The main results. For the reader’s convenience, we shall now discuss in a little moredetail the main contents of this article.

In §2 we address the issue of the undetermined exponent in (1.1) by proving that the Tamagawanumber conjecture that was formulated originally by Bloch and Kato in [3] and then subsequentlyextended by Fontaine and Perrin-Riou in [21] predicts a precise, and more-or-less explicit, formulafor nm,F . By using a result of Levine from [34] this formula can be made completely precise inthe case m = 2. In addition, results of Huber and Kings [25], Greither and the first author [14]and Flach [20] allow us to deduce the (unconditional) validity of the more precise form of (1.1)for all m and for all fields F that are abelian over Q.

The main result of §2 plays a key role in our subsequent computations and is also we feelof some independent interest. However, the arguments in §2 are technical in nature, relyingon certain 2-adic constructions and results of Kahn [27] and Rognes and Weibel [38] and ondetailed computations of the determinant modules of Galois cohomology complexes, and as


these methods are not used elsewhere in the article we invite any reader whose main interest isthe determination of explicit generators of K3-groups to simply read up to the end of §2.1 andthen pass on to §3.

In §3 we shall then introduce a useful modification p(F ) of the pre-Bloch group p(F ) thatis defined by Suslin in [42] (for in fact any infinite field), and explain how this relates both toSuslin’s construction and to (degenerate) configurations of points.

We shall also explicitly compute the torsion subgroup of the resulting modified Bloch groupB(F ) for a number field F and of the corresponding variant of the second exterior power of F ∗

in a way that is useful for implementation purposes. In particular, we show that B(F ) is torsion-free if F is an imaginary quadratic field and hence much more amenable to our computationalapproach than is Suslin’s original construction.

In this section we shall also use a result of the first three authors in [15] to construct a canonicalhomomorphism ψF from B(F ) to K3(F )ind

tf , the indecomposable K3-group of F modulo torsion.In the case that F is a number field, we are led, on the basis of extensive numerical evidence,to conjecture that ψF is bijective. This gives new insight into the long-standing questionsconcerning the precise connection between Suslin’s construction and the earlier construction ofBloch in [4] and whether Bloch’s construction accounts for all of the group K3(F )ind

tf .In this regard, it is also of interest to note that our approach to relating configurations of

points to the pre-Bloch group differs slightly, but crucially, from that described by Goncharovin [23, §3]. In particular, by these means we are able to make computations without havingto ignore torsion of exponent two, as is necessary in the approach of Goncharov. (We remarkthat this ostensibly minor improvement is in fact essential to our approach to compute explicitgenerating elements of K3-groups.)

In the remainder of the article we specialise to consider the case of an imaginary quadraticfield k.

In this case we shall firstly, in §4, invoke a tessellation of hyperbolic 3-space H3, based onperfect forms, to construct an explicit element βgeo of the group B(k) defined in §3.

The polyhedral reduction theory for GL2(Ok) that has been developed by Ash [1, Chap. II]and Koecher [30] plays a key role in this construction and Humbert’s classical formula for thevalue ζk(2) in terms of the volume of a fundamental domain for the action of PGL2(Ok) onH3 allows us to explicitly describe the image under the Beilinson regulator map of ψk(βgeo) interms of the leading term ζ∗k(−1). By using the known validity of a completely precise form ofthe equality (1.1) for F = k and m = 2 we are then able to deduce that ψk(βgeo) generates a

subgroup of K3(k)indtf of index |K2(Ok)|.

The proof that our construction of βgeo gives a well-defined element of B(k) is lengthy, andrather technical, since it relies on a detailed study of the polytopes that arise in the tessellationconstructed in §4 and, for this reason, it is deferred to §5.

Then in §6 we shall combine results from previous sections to describe two concrete approachesto finding explicit generators of K3(k)ind

tf and the order of K2(Ok) for many imaginary quadraticnumber fields k.

The first approach is discussed in §6.1 and depends upon dividing the element ψk(βgeo) con-

structed in §4 by |K2(Ok)| in an algebraic way in order to obtain a generator of K3(k)indtf . This

method partly relies on constructing non-trivial elements in B(k) by means of an algorithmicprocess involving exceptional S-units that is described in §6.3.


The second approach is discussed in §6.2 and again relies on the algorithm described in §6.3to generate elements in B(k), and hence, via the map ψk, in K3(k)ind

tf and upon knowing thevalidity of a completely precise form of (1.1). In this case, however, we combine these aspectswith some (in practice sharp) bounds on |K2(Ok)| that are provided by [2] in order to drawalgebraic conclusions from numerical calculations, leading us to the computation of a generatorof K3(k)ind

tf (or of B(k)) as well as of |K2(Ok)| in many interesting cases. As a concrete example,we describe an explicit result for an imaginary quadratic field k for which we found that |K2(Ok)|is equal to the prime 233 (thereby verifying a conjecture from [10]).

The article then concludes with two appendices. In Appendix A we shall prove several usefulresults about finite subgroups of PGL2(Ok) of an imaginary quadratic field k that are neededin earlier arguments, but for which we could not find a suitable reference. Finally, in AppendixB we shall give details of the results of applying the geometrical construction of §4 and theapproach described in §6.1 to an imaginary quadratic field k for which |K2(Ok)| is equal to 22.

1.3. Notations and conventions. As a general convention we use F to denote an arbitraryfield (assumed in places to be infinite), F to denote a number field and k to denote an imaginaryquadratic field (or, rarely, Q).

For a number field F we write OF for its ring of integers, DF for its discriminant, and r1(F )and r2(F ) for the number of its real and complex places respectively.

For an imaginary quadratic field k we set

ω = ωk :=

{√Dk/4 if Dk ≡ 0 mod 4,

(1 +√Dk)/2 if Dk ≡ 1 mod 4

so that k = Q(ω) and Ok = Z[ω].For an abelian group M we write Mtor for its torsion subgroup and Mtf for the quotient group

M/Mtor. The cardinality of a finite set X will be denoted by |X|.

1.4. Acknowledgements. We are grateful to the Irish Research Council for funding a work stayof the fourth author in Amsterdam, and to the De Brun Centre for Computational HomologicalAlgebra for funding a work stay of the second author in Galway. The initial research for thispaper was strongly propelled by these two meetings of the authors.

We also thank Mathematisches Forschungsinstitut Oberwolfach for hosting the last four au-thors to continue this work as part of their Research in Pairs program.

We would like to thank Philippe Elbaz-Vincent for useful conversations.The fifth author was partially supported by NSA grant H98230-15-1-0228. This manuscript

is submitted for publication with the understanding that the United States government is au-thorized to produce and distribute reprints.

2. The conjectures of Lichtenbaum and of Bloch and Kato

It has long been known that the validity of (1.1) follows from that of the conjecture originallyformulated by Bloch and Kato in [3] and then reformulated and extended by Fontaine in [22]and by Fontaine and Perrin-Riou in [21] (see Remark 2.8 below).

However, for the main purpose of this article, it is essential that one knows not just the validityof (1.1) but also an explicit value of the exponent nm,F for the number field F .


In this section we shall therefore derive an essentially precise formula for nm,F from theassumed validity of the above conjecture of Bloch and Kato.

If M is a Zp-module, then we identify Mtf with its image in Qp ·M := Qp ⊗Zp M and forany homomorphism of Zp-modules θ : M → N we write θtf for the induced homomorphismMtf → Ntf .

We write D(Z2) for the derived category of Z2-modules and Dperf(Z2) for the full triangu-lated subcategory of D(Z2) comprising complexes that are isomorphic (in D(Z2)) to a boundedcomplex of finitely generated modules.

2.1. Statement of the main result. Throughout this section, F denotes a number field.

2.1.1. We first review Borel’s Theorem.For each subring Λ of R and each integer a, we write Λ(a) for the subset (2πi)a · Λ of C.We then fix a natural number m and recall that Beilinson’s regulator map

regm : K2m−1(C)→ R(m− 1)

is compatible with the natural actions of complex conjugation on K2m−1(C) and R(m− 1).For each embedding σ : F → C, we consider the composite homomorphism

regm,σ : K2m−1(OF )σ∗→ K2m−1(C)

regm−→ R(m− 1)

where σ∗ denotes the induced map on K-groups.We also write ζ∗F (1−m) for the first non-zero coefficient in the Taylor expansion at s = 1−m

of the Riemann zeta function ζF (s) of F and define a natural number

dm(F ) :=

{r2(F ), if m is even,

r1(F ) + r2(F ), if m is odd.

Theorem 2.1 (Borel’s theorem). For each natural number m with m > 1 the following claimsare valid.

(i) The rank of K2m−2(OF ) is zero.(ii) The rank of K2m−1(OF ) is dm(F ).

(iii) Write regm,F for the map K2m−1(OF ) →∏σ:F→CR(m − 1) given by (regm,σ)σ. Then

the image of regm,F is a lattice in the real vector space Vm = {(cσ)σ|cσ = cσ}, and itskernel is K2m−1(OF )tor.

(iv) Normalize covolumes in Vm so that the lattice {(cσ)σ|cσ = cσ and cσ ∈ Z(m − 1)} hascovolume 1, and let Rm(F ) be the covolume of regm,F . Then ζF (s) vanishes to orderdm(F ) at s = 1−m, and there exists a non-zero rational number qm,F so that ζ∗F (1−m) =qm,k ·Rm(F ).

2.1.2. For each pair of integers i and j with j ∈ {1, 2} and i ≥ j and each prime p there existcanonical ‘Chern class’ homomorphisms of Zp-modules

(2.2) K2i−j(OF )⊗ Zp → Hj(OF [1/p],Zp(i)).

The first such homomorphism cSF,i,j,p was constructed using higher Chern class maps by Soule

in [40] (with additional details in the case p = 2 being provided by Weibel in [48]), a secondcDFF,i,j,p was constructed using etale K-theory by Dwyer-Friedlander in [18] and, in the case p = 2,

there is a third variant, introduced independently by Kahn [27] and by Rognes and Weibel [38],


that will play a key role in later arguments. In each case the maps are natural in OF and areknown to have finite kernels and cokernels (see the discussion in §2.6 for the case p = 2) andhence to induce isomorphisms of the associated Qp-spaces.

Since the prime p = 2 will be of most interest to us we usually abbreviate the maps cSF,i,j,2

and cDFF,i,j,2 to cS

F,i,j and cDFF,i,j respectively.

We can now state the main result of this section concerning the undetermined rational numberqm,F that occurs in Theorem 2.1(iv).

Theorem 2.3. Fix an integer m with m > 1. Then the Bloch-Kato Conjecture is valid for themotive h0(Spec(F ))(1−m) if and only if one has

(2.4) qm,F = (−1)sm(F )2r2(F )+tm(F ) · |K2m−2(OF )||K2m−1(OF )tor|

.

Here we set

sm(F ) :=

{[F : Q]m2 − r2(F ), if m is even,

[F : Q]m−12 , if m is odd,

and tm(F ) := r1(F ) · t1m(F ) + t2m(F ) with

t1m(F ) :=

−1, if m ≡ 1 (mod 4)

−2, if m ≡ 3 (mod 4)

1, otherwise,

and t2m(F ) the integer which satisfies

(2.5) 2t2m(F ) := |cok(cS

F,m,1,tf)|·2−am(F ) ≡ detQ2((Q2 · cSF,m,1) ◦ (Q2 · cDF

F,m,1)−1) (mod Z×2 )

where am(F ) is 0 except possibly when both m ≡ 3 (mod 4) and r1(F ) > 0 in which case it is aninteger satisfying 0 ≤ am(F ) < r1(F ).

Remark 2.6. The main result of Burgos Gil’s book [11] implies that the m-th Borel regulator of

F is equal to 2dm(F ) ·Rm(F ) and so (2.4) leads directly to a more precise form of the conjecturalformula for ζ∗F (1−m) in terms of Borel’s regulator that is given by Lichtenbaum in [35].

Remark 2.7. The proof of Lemma 2.25(ii) below gives a closed formula for the integer am(F )that occurs in Theorem 2.3 (and see also Remark 2.26 in this regard). In [34, Th. 4.5] Levineshows that cS

F,2,1, and hence also cSF,2,1,tf , is surjective so that t22(F ) = 0. In general, it is

important for the sort of numerical computations that we make subsequently to give an explicitupper bound on t2m(F ), and hence also on tm(F ), and such a bound follows directly from Theorem2.27 below.

Remark 2.8. Up to sign and an undetermined power of 2, the result of Theorem 2.3 is provedby Huber and Kings in [25, Th. 1.4.1] under the assumption that cS

F,m,j,p is bijective for allodd primes p, as is conjectured by Quillen and Lichtenbaum. In addition, Suslin has shown thelatter conjecture is a consequence of the conjecture of Bloch and Kato relating Milnor K-theoryto etale cohomology and, following fundamental work of Voevodsky and Rost, Weibel completedthe proof of the Bloch-Kato Conjecture in [47]. Our contribution to the proof of Theorem 2.3 isthus concerned solely with specifying the sign (which is straightforward) and the precise powerof 2 that should occur.


If F is an abelian field, then the Bloch-Kato conjecture for h0(Spec(F ))(1 −m) is known tobe valid: up to the 2-primary part, this was verified indepedendently by Huber and Kings in[25] and by Greither and the first author in [14] and the 2-primary component was subsequentlyresolved by Flach in [20]. Theorem 2.3 thus leads directly to the following result.

Corollary 2.9. The formula (2.4) is unconditionally valid if F is an abelian field.

Remark 2.10. Independently of connections to the Bloch-Kato conjecture, the validity of (1.1),but not (2.4), for abelian fields F was first established (modulo an uncorrected error concerningthe inclusion of Euler factors in the formulation of (1.1)) by Kolster, Nguyen Quang Do andFleckinger in [31]. Their general approach also provided motivation for the subsequent work ofHuber and Kings in [25].

Remark 2.11. In the special case that F is equal to an imaginary quadratic field k, Corollary 2.9combines with Theorem 2.1(iv), Remark 2.7 and Example 3.1 below to imply that the equality

(2.12) ζ∗k(−1) = −2|K2(Ok)|

24R2(k)

is unconditionally valid.

2.2. The proof of Theorem 2.3: a first reduction. In the sequel we abbreviate the non-negative integers r1(F ), r2(F ) and dm(F ) to r1, r2 and dm respectively.

The functional equation of ζF (s) then has the form

(2.13) ζF (1− s) =2r2 · π[F :Q]/2

|DF |1/2

(|DF |π[F :Q]

)s( Γ(s)

Γ(1− s)

)r2 ( Γ(s/2)

Γ((1− s)/2)

)r1· ζF (s).

where Γ(s) is the Gamma function.Since m is in the region of convergence of ζF (s) the value ζF (m) is a strictly positive real

number. In addition, the function Γ(s) is analytic and strictly positive for s > 0, is analytic,

non-zero and of sign (−1)12−n at each strictly negative half-integer n and has a simple pole at

each strictly negative integer n with residue of sign (−1)n.Given these facts, the above functional equation implies directly that ζF (s) vanishes at s =

1 −m to order dm (as claimed in Theorem 2.1(iv)) and that its leading term at this point has

sign equal to (−1)[F :Q]m2−r2 if m is even and to (−1)[F :Q]m−1

2 if m is odd, as per the explicitformula (2.4).

In addition, following Remark 2.8, we can (and will) assume that ζ∗F (1 − m)/Rm(F ) is arational number.

To prove Theorem 2.3 it is thus enough to show that the 2-adic component of the Bloch-KatoConjecture for h0(Spec(F ))(1 − m) is valid if and only if the 2-adic valuation of the rationalnumber ζ∗F (1−m)/Rm(F ) is as implied by (2.4).

After several preliminary steps, this fact will be proved in §2.5.

2.3. The role of Chern class maps. Regarding F as fixed we set

Km,j := K2m−j(OF )⊗ Z2 and Hjm := Hj(OF [1/2],Z2(m))

for each strictly positive integer m and each j ∈ {1, 2}. We also set

Ym := H0(GC/R,∏F→C

Z2(m− 1))


where GC/R acts diagonally on the product via its natural action on Z2(m − 1) and via post-composition on the set of embeddings F → C.

We recall that in [27] Kahn uses the Bloch-Lichtenbaum-Friedlander-Suslin-Voevodsky spec-tral sequences to construct for each pair of integers i and j with j ∈ {1, 2} and i ≥ j a homo-morphism of Z2-modules cK

i,j = cKF,i,j of the form (2.2).

We write RΓc(OF [1/2],Z2(1−m)) for the compactly supported etale cohomology of Z2(1−m)on Spec(OF [1/2]) (as defined, for example, in [12, (3)]).

We recall that this complex, and hence also its (shifted) linear dual

C•m := RHomZ2(RΓc(OF [1/2],Z2(1−m)),Z2[−2])

belongs to the category Dperf(Z2).In the sequel we shall write D(−) for the Grothendieck-Knudsen-Mumford determinant functor

on Dperf(Z2), as constructed in [29]. (Note however that, since Z2 is local, one does not lose anysignificant information by (suppressing gradings and) regarding the values of D(−) as free rankone Z2-modules and so this is what we do.)

Proposition 2.14. The map cKm,1 combines with the Artin-Verdier duality theorem to induce a

canonical identification of Q2-spaces

(2.15) Q2 ·D(C•m) = Q2 · ((∧dm

Z2

Km,1)⊗Z2 HomZ2(∧dm

Z2

· Ym,Z2))

with respect to which D(C•m) is equal to

(2.16) (2r1)t1m(F ) |Km,2|

|(Km,1)tor|· (∧dm

Z2

Km,1,tf)⊗Z2 HomZ2(∧dm

Z2

Ym,Z2)

where the integer t1m(F ) is as defined in Theorem 2.3.

Proof. We abbreviate cKm,j to cj and set bm := r1 if m is odd and bm := 0 if m is even.

Then, a straightforward computation of determinants shows that

D(C•m) =D(H0(C•m)[0])⊗D(H1(C•m)[−1]) = D(H1m[0])⊗D(H2

m[−1])⊗ 2−bmD(Ym[−1])

=(D(ker(c1)[0])−1 ⊗D(Km,1[0])⊗D(cok(c1)[0]))

⊗ (D(ker(c2)[−1])−1 ⊗D(Km,2[−1])⊗D(cok(c2)[−1]))⊗ 2−bmD(Ym[−1])

=2−bm|ker(c1)|·|cok(c2)||ker(c2)|·|cok(c1)|

|Km,2||(Km,1)tor|

· (∧dm

Z2


Z2

Ym,Z2).

Here the first equality is valid because C•m is acyclic outside degrees zero and one, the secondfollows from the descriptions of Lemma 2.19 below, the third is induced by the tautological exact

sequences 0 → ker(cj) → Km,jcj−→ Hj

m → cok(cj) → 0 for j = 1, 2 and the last equality followsfrom the fact that for any finitely generated Z2-module M and integer a the lattice D(M [a]) is

equal to (|Mtor|)−1·∧b

Z2Mtf with b = dimQ2(Q2·M) if a is even and to |Mtor|·HomZ2(

∧bZ2Mtf ,Z2)

if a is odd.


It thus suffices to show the product of 2−bm and |ker(c1)|·|cok(c2)|/(|ker(c2)|·|cok(c1)|) is

(2r1)t1m(F ) and this follows by explicit computation using the fact that [27, Th. 1] implies

(2.17)

|ker(cj)|= |coker(cj)|= 1, if 2m− j ≡ 0, 1, 2, 7 (mod 8)

|ker(cj)|= 2r1 , |coker(cj)|= 1, if 2m− j ≡ 3 (mod 8)

|ker(cj)|= 2r1,4 , |coker(cj)|= 1, if 2m− j ≡ 4 (mod 8)

|ker(cj)|= 1, |coker(cj)|= 2r1,5 , if 2m− j ≡ 5 (mod 8)

|ker(cj)|= 1, |coker(cj)|= 2r1 , if 2m− j ≡ 6 (mod 8)

where the integers r1,4 and r1,5 are zero unless r1 > 0 in which case one has r1,4 ≥ 0, r1,5 > 0and r1,4 + r1,5 = r1. �

Remark 2.18. In [38] Rognes and Weibel use a slightly different approach to Kahn to constructmaps of the form cK

i,j . The results obtained in loc. cit. can be used to give an alternative proofof Proposition 2.14.

In the sequel we write Σ∞, ΣR and ΣC for the sets of archimedean, real archimedean andcomplex archimedean places of F respectively.

Lemma 2.19. The Artin-Verdier duality theorem induces the following identifications.

(i) H0(C•m) = H1(OF [1/2],Z2(m)).(ii) H1(C•m)tor = H2(OF [1/2],Z2(m)).

(iii) H1(C•m)tf is the submodule⊕w∈ΣR

2 ·H0(GC/R,Z2(m− 1))⊕⊕w∈ΣC

H0(GC/R,Z2(m− 1) · σw ⊕ Z2(m− 1) · σw)

of

Ym =⊕w∈ΣR

H0(GC/R,Z2(m− 1))⊕⊕w∈ΣC

H0(GC/R,Z2(m− 1) · σw ⊕ Z2(m− 1) · σw),

where for each place w ∈ ΣC we choose a corresponding embedding σw : F → C and writeσw for its complex conjugate.

Proof. For each w in Σ∞ we write RΓTate(Fw,Z2(1−m)) for the standard complete resolutionof Z2(1−m) that computes Tate cohomology over Fw and RΓ∆(Fw,Z2(1−m)) for the mappingfibre of the natural morphism RΓ(Fw,Z2(1−m))→ RΓTate(Fw,Z2(1−m)).

We recall from, for example, [12, Prop. 4.1], that the Artin-Verdier Duality Theorem givesan exact triangle in Dperf(Z2) of the form

(2.20) C•m →⊕v∈Σ∞

RHomZ2(RΓ∆(Fw,Z2(1−m)),Z2[−1])→ RΓ(OF [1/2],Z2(m))[2]→ C•m[1].

In addition, explicit computation shows that RHomZ2(RΓ∆(Fw,Z2(1−m)),Z2[−1]) is repre-sented by the complex{

Z2(m− 1)[−1], if w ∈ ΣC

(Z2(m− 1)δ0m−→ Z2(m− 1)

δ1m−→ Z2(m− 1)

δ0m−→ · · ·)[−1], if w ∈ ΣR,

with δim := 1− (−1)iτ for i = 0, 1 where τ denotes complex conjugation.


Given this description, and the fact that H2(OF [1/2],Z2(m)) is finite, the long exact coho-mology sequence of (2.20) leads directly to the identifications in claims (i) and (ii) and also givesan exact sequence of Z2-modules as in the upper row of the commutative diagram

0 H1(C•m)tf⊕

w∈Σ∞H0(Fw,Z2(m− 1)) H3(OF [1/2],Z2(m))

⊕w∈Σ∞

H3(Fw,Z2(m))⊕

w∈Σ∞H3(Fw,Z2(m))

θ ∼=

in which the right hand vertical isomorphism is the canonical isomorphism and θ is defined tomake the square commute.

Since H3(Fw,Z2(m)) is isomorphic to Z2/2Z2 if w ∈ ΣR and m is odd and vanishes in allother cases and θ is known to respect the direct sum decompositions of its source and target(see the proof of [12, Lem. 18]), this diagram implies the description of H1(C•m)tf given in claim(iii). �

Remark 2.21. The proof of Lemma 2.19 also shows that [25, Rem. following Prop. 1.2.10]requires modification. Specifically, and in terms of the notation of loc. cit., for the givenstatement to be true the term det(T2(r)+) must be replaced by |H0(R, T2(r))|·det(T2(r)+) rather

than by det(H0(R, T2(r))), as asserted at present.

2.4. Pairings on Betti cohomology and Beilinson’s regulator. We start with an obser-vation concerning a natural perfect pairing on Betti-cohomology.

To do this we write ER for the set of embeddings F → C that factor through R ⊂ C and setEC := Hom(F,C) \ ER. For each integer a we set Wa,R :=

∏ER(2πi)aZ and Wa,C :=

∏EC(2πi)aZ

and endow the direct sum Wa := Wa,R ⊕Wa,C with the diagonal action of GC/R that uses its

natural action on (2πi)m−1Z and post-composition on the embeddings F → C.We write τ for the non-trivial element of GC/R and for each GC/R-module M we use M± to

denote the submodule comprising elements upon which τ acts as multiplication by ±1.Then the perfect pairing

(Q⊗Z Wa)× (Q⊗Z W1−a)→ (2πi)Qthat sends each element ((cσ), (c′τ )) to

∑σ cσc

′σ restricts to induce an identification

HomZ(W+a , (2πi)Z) = W−1−a,R ⊕ (W1−a,C/(1 + τ)W1−a,C)

and hence alsoHomZ(W+

a ,Z) = W+−a,R ⊕ (W−a,C/(1− τ)W−a,C).

In particular, after identifying Q⊗ (W−a,C/(1− τ)W−a,C) and Q⊗W+−a,C in the natural way,

one obtains an isomorphism

(2.22) Q⊗W+−a∼= HomQ(Q⊗W+

a ,Q)

that identifies W+−a with a sublattice of HomZ(W+

a ,Z) in such a way that

(2.23) |HomZ(W+a ,Z)/W+

−a|= 2r2 .

Next, for each natural number m with m > 1 we consider the composite homomorphism

regm : K2m−1(Γ)→ H1D(Spec(Γ), (2πi)R) = Γ/(2πi)mR→ (2πi)m−1R


where the first arrow is the Beilinson regulator map and the second is the isomorphism inducedby the decomposition Γ = (2πi)mR⊕ (2πi)m−1R. We then write

(2.24) βm : R⊗K2m−1(OF )→ R⊗W+m−1

∼= HomR(R⊗W+1−m,R)

for the composite homomorphism of R[G]-modules where the first map is induced by the com-posites of regm with the maps K2m−1(F ) → K2m−1(C) that are induced by each embeddingF → C and the second by the isomorphism (2.22) with a = 1−m.

The map βm is bijective and we write βm,∗ for the induced isomorphism of R-spaces

R⊗ (∧dm

ZK2m−1(OF ))⊗HomZ(

∧dm

ZHomZ(W+

1−m,Z),Z)→ R

that is induced by βm.

2.5. Completion of the proof of Theorem 2.3. After making explicit the formulation of[13, Conj. 1] (which originates with Fontaine and Perrin-Riou [21, Prop. III.3.2.5]) and theconstruction of [13, Lem. 18], one finds that the Bloch-Kato Conjecture for h0(Spec(F ))(1−m)uses the composite map βm in (2.24) rather than simply the first map that occurs in its definition.

In addition, if one fixes a topological generator η of Z2(m − 1), then by mapping η to theelement of HomQ((2πi)1−mQ,Q) that sends (2πi)1−m to 1 one obtains an identification of Ymwith Z2 ⊗HomZ(W+

1−m,Z).Given these observations, the discussion of [25, §1.2] shows the Bloch-Kato Conjecture asserts

that if one fixes an identification of C with C2, then ζ∗F (1 − m) is a generator over Z2 of theimage of D(C•m) under the composite isomorphism

C2 ·D(C•m) ∼= C2 · ((∧dm

Z2

Km,1)⊗Z2 HomZ2(∧dm

Z2

· Ym,Z2)) ∼= C2.

Here the first map differs from the scalar extension of (2.15) only in that, in its construction,the map cS

F,m,1 is used in place of cKF,m,1 and the second isomorphism is C2 ⊗R βm,∗.

In the sequel we write detQ2(α) for the determinant of an automorphism of a finite dimensionalQ2-vector space.

Then, by combining the above observations together with the result of Proposition 2.14 onefinds that the Bloch-Kato Conjecture predicts the image under C2 ⊗R βm,∗ of the lattice (2.16)to be generated over Z2 by the product ζ∗F (1−m) · detQ2((Q2 · cS

F,m,1) ◦ (Q2 · cKF,m,1)−1)−1.

In addition, since the regulator Rm(F ) is defined with respect to the lattice W+m−1 rather than

HomZ(W+1−m,Z), the index formula (2.23) implies that

(C2 ⊗R βm,∗)((∧dm

Z2


Z2

Ym,Z2)) = 2r2 ·Rm(F ) · Z2 ⊂ C2.

To deduce the explicit formula (2.4) from these facts (by direct substitution) we now needonly note that Lemma 2.25 below implies

2am(F )|cok(cSF,m,1,tf)|−1·detQ2((Q2 · cS

F,m,1) ◦ (Q2 · cKF,m,1)−1) ∈ Z×2 .

Finally we note that the congruence which occurs in (2.5) is a direct consequence of Lemma2.25(ii) below and this then completes the proof of Theorem 2.3.

Lemma 2.25.

(i) One has detQ2((Q2 · cDFF,m,1) ◦ (Q2 · cK

F,m,1)−1) ∈ Z×2 .


(ii) Define a non-negative integer am(F ) by the equality 2am(F ) = |cok(cKF,m,1,tf)|. Then

2am(F )|cok(cSF,m,1,tf)|−1·detQ2((Q2 · cS

F,m,1) ◦ (Q2 · cDFF,m,1)−1) ∈ Z×2

and am(F ) is equal to 0 except possibly when both m ≡ 3 (mod 4) and r1 > 0 in whichcase one has am(F ) = r1,5 − 1 ≥ 0.

Proof. Set F ′ := F (√−1) and ∆ := GF ′/F . With θF ′ denoting either cS

F ′,i,1, cDFF ′,i,1 or cK

F ′,i,1we write θF for the corresponding map defined at the level of F . Then there is a commutativediagram

K2m−1(OF ′)⊗ Z2 H1(OF ′ [1/2],Z2(m))

K2m−1(OF )⊗ Z2 H1(OF [1/2],Z2(m))

θF ′

θF

in which the left hand vertical arrow is induced by the inclusion OF ⊆ OF ′ and the right handvertical arrow is the natural restriction map. In addition, since θF ′ is natural with respect toautomorphisms of F ′ it is ∆-equivariant and hence, upon taking ∆-fixed points, and noting thatthe right hand vertical arrow identifies H1(OF [1/2],Q2(m)) with H0(∆, H1(OF ′ [1/2],Q2(m))),the diagram identifies Q2 ·H0(∆, θF ′) with Q2 · θF .

Now, in [18, Th. 8.7 and Rem. 8.8] Dwyer and Friedlander show that the map cDFF ′,m,1 is

surjective and hence, since it maps between finitely generated Z2-modules of the same rank,the induced map cDF

F ′,m,1,tf is bijective. In addition, since r1(F ′) = 0, the description (2.17)

implies cKF ′,m,1, and hence also cK

F ′,m,1,tf , is bijective. One thus has cDFF ′,m,1,tf = ϕ ◦ cK

F ′,m,1,tf for

an automorphism ϕ of the Z2[∆]-lattice ΞF ′ := Hj(OF ′ [1/2],Z2(m))tf . Restricting to ∆-fixedpoints this implies that

detQ2((Q2 · cDFF,m,1) ◦ (Q2 · cK

F,m,1)−1) =detQ2(H0(∆,Q2 · cDFF ′,m,1) ◦H0(∆,Q2 · cK

F ′,m,1)−1)

=detQ2(H0(∆,Q2 · ϕ))

and this belongs to Z×2 since H0(∆, ϕ) is an automorphism of the Z2-lattice H0(∆,ΞF ′). Thisproves claim (i).

In view of claim (i) it suffices to prove claim (ii) with cDFF,m,1 replaced by cK

F,m,1. By (2.17) one

also knows cKF,m,1, and hence also cK

F,m,1,tf , is surjective except possibly if m ≡ 3 (mod 4) and

r1 > 0. In the latter case, the Z2-module Km,1 is torsion-free (by [38, Th. 0.6]) and, since r1 > 0and m is odd, it is straightforward to check that |H1

m,tor|= 2 (see, for example, [38, Props.

1.8 and 1.9(b)]). In this case therefore, the computation (2.17) implies that |cok(cKF,m,1,tf)|=

2−1 · |cok(cKF,m,1)|= 2r1,5−1.

Given this, and the explicit definition of am(F ), claim (ii) follows directly from the fact that,when computed with respect to any fixed Z2-bases of (K2m−1(OF )⊗Z2)tf and ΞF , the determi-

nants of cKF,m,1,tf and cS

F,m,1,tf are generators of 2am(F ) ·Z2 and |cok(cSF,m,1,tf)|·Z2 respectively. �

Remark 2.26. In [27, just after Th. 1] Kahn explicitly asks whether if m ≡ 3 (mod 4) andr1 > 0 the integer r1,5 in (2.17) is always equal to 1 and points out that this amounts to askingwhether, in this case, the image of H1(OF [1/2],Z2(m)) in H1(OF [1/2],Z/2) ⊂ F×/(F×)2 iscontained in the subgroup generated by the classes of ±1 and the set of totally positive elementsof F×? If the answer to this question is affirmative, then in all cases one has am(F ) = 0.


2.6. An upper bound for t2m(F ). We henceforth fix a natural number m with m > 1 and foreach number field E abbreviate Soule’s 2-adic Chern class map

cSE,m,1,2 : K2m−1(OE)⊗ Z2 → H1(OE [1/2],Z2(m))

to cE . We also set E′ := E(√−1).

The computation made in §2.3 relied crucially on the fact that the cokernel of cF is finite.However, whilst this fact is well-known to experts, we were not able to locate a proof in theliterature (and also see Remark 2.30 below).

In addition, and as already discussed in Remark 2.7, for the purposes of numerical computa-tions, one must not only know that cok(cF ) is finite but also have a computable upper bound

for cok(cF,tf) = 2t2m(F ).

In this section we address these issues by proving the following result.

Theorem 2.27. cok(cF,tf) is finite and its cardinality divides

[F ′ : F ]dm(F )((m− 1)! )dm(F ′)|K2m−2(OF ′)|.

As preparation for the proof of Theorem 2.27 we first consider universal norm subgroups inetale cohomology. To do this we write F ′∞ for the cyclotomic Z2-extension of F ′ and Fn foreach non-negative integer n for the unique subfield of F ′∞ of degree 2n over F ′. We also setΓ := GF ′∞/F ′ and write Λ for the Iwasawa algebra Z2[[Γ]]. For each Λ-module N and integer awe write N(a) for the Λ-module N ⊗Z2 Z2(a) upon which Γ acts diagonally.

For each finite extension E of F ′ we abbreviate OE [1/2] to O′E . We then define the ‘universalnorm’ subgroup H1

∞(O′F ′ ,Z2(m)) of H1(O′F ′ ,Z2(m)) to be the image of the natural projec-tion map lim←−nH

1(O′F ′n ,Z2(m)) → H1(O′F ′ ,Z2(m)) where the limit is taken with respect to the

natural corestriction maps.

Proposition 2.28. The index of H1∞(O′F ′ ,Z2(m)) in H1(O′F ′ ,Z2(m)) divides |K2m−2(OF ′)|.

Proof. We write C•∞ for the object of the derived category of perfect complexes of Λ-modulesthat is obtained as the inverse limit of the complexes RΓ(O′F ′n ,Z2(m)) with respect to the natural

projection morphisms

RΓ(O′F ′n+1,Z2(m))→ Zp[Γn]⊗L

Zp[Γn+1] RΓ(O′F ′n+1,Z2(m)) ∼= RΓ(O′F ′n ,Z2(m)).

We recall that there is a natural isomorphism Z2 ⊗LΛ C

•∞∼= RΓ(O′F ′ ,Z2(m)) in Dperf(Z2) and

that this induces a natural short exact sequence of Z2-modules

(2.29) 0→ Z2 ⊗Z2[[Γ]] H1(C•∞)

π−→ H1(O′F ′ ,Z2(m))→ H0(Γ, H2(C•∞))→ 0.

In addition, in each degree i one has H i(C•∞) = lim←−nHi(O′F ′n ,Z2(m)), where the limits are

taken with respect to the natural corestriction maps, and so im(π) = H1∞(O′F ′ ,Z2(m)) and the

Λ-module H2(C•∞) is isomorphic to (lim←−nHi(O′F ′n ,Z2(1)))(m− 1).

Now for each n class field theory identifies H2(O′F ′n ,Z2(1))tor with the ideal class group

Pic(O′F ′n) of O′F ′n and H2(O′F ′n ,Z2(1)) with a submodule of the free Z2-module on the set of

places of F ′n that are either archimedean or 2-adic and hence, upon passing to the limit over n


and then taking Γ-invariants, gives rise to an exact sequence of Z2-modules

0 → H0(Γ, X ′∞(m − 1)) → H0(Γ, H2(C•∞)) →⊕

v∈Σ2∪Σ∞(F ′)

H0(Γ,Z2[[Γ/Γv]](m − 1))

where X ′∞ is the Galois group of the maximal unramified pro-2 extension of F∞ in which all 2-adic places split completely, Σ2 the set of 2-adic places of F ′ and Γv the decomposition subgroupof v in Γ. Since m > 1 it is also clear that each module H0(Γ,Z2[[Γ/Γv]](m− 1)) vanishes andhence that the sequence implies H0(Γ, X ′∞(m− 1)) = H0(Γ, H2(C•∞)).

In view of the exact sequence (2.29) we are thus reduced to showing that H0(Γ, X ′∞(m− 1))is finite and of order dividing |K2m−2(OF ′)|.

Next we recall that, by a standard ‘Herbrand Quotient’ argument in Iwasawa theory (see, forexample, [45, Exer. 13.12]), if a finitely generated Λ-module N is such that H0(Γ, N) is finite,then H0(Γ, N) is both finite and of order at most |H0(Γ, N)|. In addition, since F ′ is totallyimaginary, the argument of Schneider in [39, §6, Lem. 1] (see also the discussion of Le Floc’h,Movahhedi and Nguyen Quang Do in [33, just before Lem. 1.2]) shows that H0(Γ, X ′∞(m− 1))is naturally isomorphic to the ‘etale wild kernel’

WKet2m−2(F ′) := ker(H2(O′F ′ ,Z2(m))→

⊕v∈Σ2(F ′)∪Σ∞(F ′)

H2(F ′w,Z2(m)))

of F ′, where the arrow denotes the natural diagonal localisation map.Hence, to deduce the claimed result, we need only recall that, as F ′ is totally imaginary,

the group H2(O′F ′ ,Z2(m)) is naturally isomorphic to the finite group K2m−2(OF ′) ⊗Z Z2 (as aconsequence of (2.17)). �

Turning now to the proof of Theorem 2.27, we consider for each n the following diagram

K2m−1(O′F ′n ,Z/2n) H1(O′F ′n , (Z/2

n)(m)) H1(F ′n, (Z/2n)(m))

K2m−1(O′F ′ ,Z/2n) H1(O′F ′ , (Z/2n)(m)) H1(F ′, (Z/2n)(m)) .

m·cF ′n,2n ιn

m·cF ′,2n ι

Here we write cE,2n for the Chern class maps K2m−1(O′E ,Z/2n)→ H1(O′E , (Z/2n)(m)) of Soule,as discussed by Weibel in [48], the arrows ιn and ι are the natural inflation maps, the left handvertical arrow is the natural transfer map and the remaining vertical arrows are the naturalcorestriction maps. In particular, the results of [48, Prop. 2.1.1 and 4.4] combine to imply thatthe outer rectangle of this diagram commutes, and hence since the maps ιn and ι are injective,that the first square also commutes.

Since the first square is compatible with change of n in the natural way we may then pass tothe inverse limit over n to obtain a commutative diagram

lim←−nK2m−1(O′F ′n ,Z/2n) lim←−nH

1(O′F ′n ,Z2(m))

K2m−1(O′F ′)⊗Z Z2 H1(O′F ′ ,Z2(m)) .

(m·cF ′n,2n )n

m·cF ′


Here we use the fact that, since K2m−1(O′F ′) is finitely generated, lim←−nK2m−1(O′F ′ ,Z/2n) iden-

tifies with K2m−1(O′F ′)⊗Z Z2 in such a way that the limit (m · cF ′,2n)n is equal to m · cF ′ .Now the image of the right hand vertical arrow in this diagram is H1

∞(O′F ′ ,Z2(m)) and,as each F ′n contains all roots of unity of order 2n, from [48, Cor. 5.6] one knows that theexponent of cok((m · cF ′n,2n)n) divides m!. From the commutativity of the above diagram we

can therefore deduce im(m · cF ′) contains m! ·H1∞(O′F ′ ,Z2(m)) and hence also that im(cF ′,tf)

contains (m− 1)! ·H1∞(O′F ′ ,Z2(m))tf . This inclusion implies that |cok(cF ′,tf)| divides

|(H1(O′F ′ ,Z2(m))tf/H1∞(O′F ′ ,Z2(m))tf)|·|(H1

∞(O′F ′ ,Z2(m))tf/(m− 1)! ·H1∞(O′F ′ ,Z2(m))tf)|

and hence also |(H1(O′F ′ ,Z2(m))/H1∞(O′F ′ ,Z2(m)))|·((m− 1)! )dm(F ′).

Recalling Proposition 2.28 it now follows |cok(cF ′,tf)| divides ((m − 1)! )dm(F ′)|K2m−2(OF ′)|,as claimed by Theorem 2.27 in the case F ′ = F .

To deduce the general case of Theorem 2.27 we assume F ′ 6= F , write τ for the uniquenon-trivial element of ∆ and note [48, Prop. 4.4] implies there is a commutative diagram

K2m−1(O′F ′)⊗Z Z2 H1(O′F ′ ,Z2(m))tf

K2m−1(O′F )⊗Z Z2 H1(O′F ,Z2(m))tf

cF ′,tf

T 1∆

T 2∆

cF,tf

where the maps T i∆ are induced by the respective actions of 1 + τ ∈ Z2[∆]. The commutativityof this diagram implies the index of cF,tf(im(T 1

∆)) in im(T 2∆) divides |cok(cF ′,tf)|. Thus, since

H1(O′F ,Z2(m))tf identifies with a (finite index) subgroup of H0(∆, H1(O′F ′ ,Z2(m))tf), to deduceTheorem 2.27 from the special case F = F ′ it is enough to show that the Tate cohomology group

H0(∆, H1(O′F ′ ,Z2(m))tf) := H0(∆, H1(O′F ′ ,Z2(m))tf)/im(T 2∆)

has cardinality dividing 2dm(F ) and this is true because this quotient has exponent dividing|∆|= 2 and the Z2-lattice H0(∆, H1(O′F ′ ,Z2(m))tf) has rank dm(F ).

This completes the proof of Theorem 2.27.

Remark 2.30. The argument of Huber and Wildeshaus in [26, Th. B.4.8 and Lem. B.4.7] aimsto show, amongst other things, that cok(cS

F,m,1,2) is finite. However, this argument uses in a key

way results of Dwyer and Friedlander from [18, Th. 8.7 and Rem. 8.8] that relate to cDFF,m,1,2

rather than cSF,m,1,2. To complete this argument one would thus need to investigate the relation

between cok(cDFF,m,1,2) and cok(cS

F,m,1,2).

3. K-theory, wedge complexes, and configurations of points

Let F be an arbitrary field. Then it is well-known by work of Bloch and (subsequently) Suslinthat K3(F) is closely related to the Bloch group B(F) (as defined in §3.3.4 below). However,the group B(F) often contains non-trivial elements of finite order and so can be difficult for thepurposes of explicit computation. With this in mind, in this section we shall introduce a slightvariant B(F ) of B(F ) on which we have better control.


We shall also construct a canonical homomorphism ψF from B(F) to K3(F)indtf (see Theorem

3.20) and are motivated to conjecture, on the basis of extensive computational evidence, thatψF is bijective if F is a number field (see Conjecture 3.33). We note, in particular, that theseobservations provide the first concrete evidence to suggest both that the groups defined by Suslinin terms of group homology and by Bloch in terms of relative K-theory should be related ina very natural way, and also that Bloch’s group should account for all of the indecomposableK3-group, at least modulo torsion (cf. Remark 3.34).

If F is imaginary quadratic, then the groups B(F) and K3(F)indtf are both isomorphic to Z (see

Corollary 3.29 for B(F) and we shall later use ψF to reduce the problem of finding a generatingelement of K3(F)ind

tf to computational issues in B(F).

3.1. Towards explicit versions of K3(F) and reg2. In this section we review a result of thefirst three authors in [15].

3.1.1. We first recall some basic facts concerning the K3-groups of a general field F . To thisend we write K3(F)ind for the quotient of K3(F) by the image of the Milnor K-group KM

3 (F)of F .

We recall that if the field is a number field F , then the abelian group KM3 (F ) has exponent 1

or 2 and order 2r1(F ) (cf. [46, p.146]), so that K3(F )indtf identifies with K3(F )tf , hence is a free

abelian group of rank r2(F ) as a consequence of Theorem 2.1(ii).We further recall that for any field F the torsion subgroup of K3(F)ind is explicitly described

by Levine in [34, Cor. 4.6].

Example 3.1. If F is equal to an imaginary quadratic extension k of Q in Q, then the latterresult gives isomorphisms

K3(k)indtor ' H0(Gal(Q/k),Q(2)/Z(2)) = H0(Gal(Q/Q),Q(2)/Z(2)) = Z/24Z

where the first equality is valid because complex conjugation acts trivially on Q(2)/Z(2) and thesecond follows by explicit computation. For any such field k the abelian group K3(k)ind = K3(k)is therefore isomorphic to a direct product of the form Z× Z/24Z.

3.1.2. For an arbitrary field F we set

∧2F∗ :=F∗ ⊗Z F∗

〈(−x)⊗ x with x in F∗〉

(which, we note, differs from the usual exterior power ∧2F∗ because of the sign in the denomi-nator).

For each a and b in F∗ we write a∧b for the class of a⊗ b in ∧2F∗. Then it is easily verifiedthat the sum a∧b+ b∧a is trivial.

We next set F [ := F \ {0, 1} and write Z[F [] for the free Abelian group on F [. We considerthe homomorphism

(3.2) δ2,F :Z[F []→ ∧2F∗

that for each x in F [ sends [x] to (1− x)∧x.

We write D(z):C[ → R for the Bloch-Wigner dilogarithm. This function is defined by Blochin [4] by integrating the function log|w|·darg(1 − w) − log|1 − w|·darg(w) along any path from


a point z0 in R[ to z. We recall that, by differentiating, it is easily shown that this functionsatisfies the identities

(3.3)

D(z) +D(z−1) = 0, D(z) +D(1− z) = 0, D(z) +D(z) = 0,

D(x)−D(y) +D(yx

)−D

(1− y1− x

)+D

(1− y−1

1− x−1

)= 0,

for x, y, and z in C[ with x 6= y.We note, in particular, that the third identity here implies that the map iD from C[ to R(1)

is equivariant with respect to the natural action of complex conjugation.

The following result describes an important connection between these constructions.

Theorem 3.4 ([15, Th. 4.1]). With the above notation, the following claims are valid.

(i) There exists a homomorphism

ϕF : ker(δ2,F )→ K3(F)indtf ,

that is natural up to sign, and, after fixing a choice of sign, functorial in F .(ii) If F is a number field F , then the cokernel of ϕF is finite.

(iii) There exists a universal choice of sign such that if F is any number field, and σ:F → Cis any embedding, then the composition

regσ: ker(δ2,F ) K3(F )indtf = K3(F )tf K3(C)tf R(1)

ϕF σ reg2

is induced by sending each element [x] for x in F [ to iD(σ(x)).

3.2. Analysing the twisted wedge product. In this section we obtain explicit information

on the structure of ∧2F∗ for a general field F . With an eye towards implementation for thepurposes of numerical calculations, we pay special attention to the case that F is a number field.

3.2.1. We first consider the abstract structure of ∧2F∗.

Proposition 3.5. For a field F , we have a filtration {0} = Fil0 ⊆ Fil1 ⊆ Fil2 ⊆ Fil3 = ∧2F∗,with Fil1 the image of F∗tor ⊗F∗tor and Fil2 the image of F∗tor ⊗F∗. Then

Fil2 =F∗tor ⊗F∗

〈(−x)⊗ x with x in F∗tor〉

and there are natural isomorphisms

Fil1/Fil0 = ∧2F∗tor =F∗tor ⊗Z F∗tor

〈(−x)⊗ x with x in F∗tor〉Fil2/Fil1 ' F∗tor ⊗F∗tf

Fil3/Fil2 'F∗tf ⊗Z F∗tf

〈x⊗ x with x in F∗tf〉,

with the last two induced by the quotient maps F∗tor⊗F∗ → F∗tor⊗F∗tf and F∗⊗F∗ → F∗tf ⊗F∗tf .


Proof. By taking filtered direct limits, it suffices to prove those statements with F∗ replacedwith a finitely generated subgroup A of F∗ that contains −1. We can then obtain a splittingA ' Ator ⊕Atf and find that the quotient for A is isomorphic to

Ator ⊗Ator ⊕Ator ⊗Atf ⊕Atf ⊗Ator ⊕Atf ⊗Atf

〈((−u)⊗ u, (−u)⊗ c, c⊗ u, c⊗ c) with u in Ator and c in Atf〉.

Our claims follow for A if we prove that the intersection of

Ator ⊗Ator ⊕Ator ⊗Atf ⊕Atf ⊗Ator ⊕ 0

with the group in the denominator equals

〈((−u)⊗ u, u⊗ c, c⊗ u, 0) with u in Ator and c in Atf〉

as the latter is the product

〈(−u)⊗ u with u in Ator〉 × 〈(v ⊗ c, c⊗ v) with v in Ator and c in Atf〉 × {0} .

From the identity

(−uc)⊗ uc− (−u)⊗ u− (−c)⊗ c = u⊗ c+ c⊗ uin A× A it is clear that this intersection contains the given subgroup, so we only have to showit is not larger. For this, choose a basis b1, . . . , bs of Atf and assume that, for some integers mi,the last position in

(3.6)∑i

mi((−ui)⊗ ui, (−ui)⊗ ci, ci ⊗ ui, ci ⊗ ci)

is trivial. If ai,j is the coefficient of bj in ci, then∑

imia2i,j = 0 for each j. So each sum

∑imiai,j

is even, the sum ∑i

mi(−1)⊗ ci =∑j

∑i

miai,j(−1)⊗ bj

is trivial, and in the second position of the element in (3.6) we can replace each −ui with ui. �

Remark 3.7. Clearly, Fil1 is trivial if F has characteristic 2. It is also trivial if that characteris-tic is not equal to 2 but F∗ contains an element of order 4: if u in F∗ has order 2m with m even,

then u∧u = (−1)∧u = m(u∧u) in ∧2F∗, and gcd(m−1, 2m) = 1. Finally, if F has characteristicnot equal to 2, and F∗ does not contain an element of order 4, then by decomposing F∗tor intoits primary components, one sees that Fil1 is cyclic of order 2, generated by (−1)∧(−1).

Corollary 3.8. Let F be a number field, n the order of F ∗tor, and c1, c2, . . . in F ∗ such that theygive a basis of F ∗tf . Let m = 1 and u = 1 if n is divisible by 4, and m = 2 and u = −1 otherwise.Then the map

Z/mZ×⊕iZ/nZ×⊕i<jZ→ ∧2F ∗

(a, (bi)i, (bi,j)i,j) 7→ ua∧u+∑i

ubi∧ci +∑i<j

cmi,ji ∧cj

is an isomorphism.


Proof. There is a filtration Fil′l for l = 0, 1, 2 and 3 on the domain by taking the last 3 − lpositions to be trivial, and under the homomorphism we map Fil′l to Fill as in Proposition 3.5so it induces a homomorphism Fil′l/Fil′l−1 → Fill/Fill−1 for l = 1, 2 and 3. For l = 1 this is anisomorphism by Remark 3.7, and for l = 2 and l = 3 by Proposition 3.5. We now apply thefive-lemma. �

Remark 3.9.(i) It is not always possible to choose decompositions as in the corollary that are compatible withfield extensions (hence the formulation of Proposition 3.5 in terms of filtrations). For example,if this were possible when we extend Q to Q(

√−1), then the decomposition of 2 in Q∗ would

map to one in Q(√−1)∗ of the form (±1)α2 with α in the chosen free part for Q(

√−1)∗ because

2 maps to a square in Q(√−1)∗tf . But neither 2 nor −2 are squares in Q(

√−1)∗. However, it is

possible for F ′/F if F and F ′ have the same roots of unity: choose ci for F and c′i for F ′, thenadjust each ci with a root of unity such that the result is in the free part generated by the c′i.

(ii) One can get finitely many of the ci in the corollary by taking a basis of the free part of theS-units for a finite set S of primes of the ring of integers O of k. If one extends S to S′, thenone can add more cj in order to obtain a similar basis for the S′-units.

(iii) Given a generator u of F ∗tor of order 2l and finitely many of the ci in the corollary, thattogether generate a subgroup A of F ∗, the isomorphism of the corollary becomes explicit on the

image of ∧2A by writing everything out in terms of the generators, and using ci∧cj + cj∧ci = 0

if i 6= j, ci∧ci = (−1)∧ci, u∧ci + ci∧u = 0, as well as that u∧u is equal to (−1)∧(−1) if l is odd,and trivial if l is even.

3.2.2. Any field extension F → F ′ induces a homomorphism from ∧2F∗ to ∧2(F ′)∗. In the

next result we determine the kernel of this map in the case that F = Q and F ′ is imaginaryquadratic. This result will then play a important role for Theorem 4.7.

Lemma 3.10. Let d be a positive square-free integer, and let k = Q(√−d). If d 6= 1 then the

kernel of the map ∧2Q∗ → ∧2k∗ has order 2, with (−1)∧(−d) as non-trivial element. If d = 1

then the kernel is non-cyclic of order 4, and is generated by (−1)∧(−1) and (−1)∧2.

Proof. The given elements are clearly in the kernel (for Q(√−1) use 2 =

√−1(1 −

√−1)2 and

that√−1∧√−1 is trivial by Remark 3.7). In addition, by Proposition 3.5 (or Corollary 3.8 with

the prime numbers as ci) the elements generate a subgroup of the stated order. It is thereforeenough to check the size of the kernel.

Clearly, from the description in Proposition 3.5 the kernel is contained in Fil2 on ∧2Q. Becausethe map on the Fil1-pieces is surjective by Remark 3.7, with kernel of order 2 if d = 1 and trivialotherwise, we only have to show that the kernel for Fil2/Fil1 has order 2.

For this we use the description of Fil2/Fil1 in Proposition 3.5. If |k∗tor|= 2m then the kernelcorresponds to the kernel of the map Q∗tf/2→ k∗tf/2m given by raising to the mth power, as −1is the mth power of a generator of k∗tor. We solve am = uα2m with a in Q∗, u in k∗tor, and αin k∗, or, equivalently, a = vα2 for some v in k∗tor.

For d 6= 1, we find that a is of the form ±b2 or ±d · b2 for some b in Q∗, and for d = 1 it is ofthe form ±b2 or ±2b2. In either case, this leads to two elements in Q∗tf/2 that are in the kernel,as required. �


3.3. Configurations of points, and a modified Bloch group. In order to be able to usethe result of Theorem 3.4 in the geometric construction of elements in the indecomposable K3-groups of imaginary quadratic fields that we shall make in §4, it is convenient to make technicalmodifications of well-known constructions of Suslin from [42] and of Goncharov from [23, p. 73].As a result, we shall be able to be more precise about torsion in the resulting Bloch groups andsome of the homomorphisms involved.

However, in order to be able to take finite non-trivial torsion in stabilisers of points in P1F

into account, and to be able to work with groups like PGL2(F) instead of GL2(F) whenevernecessary, we are forced to work in somewhat greater generality.

3.3.1. Let F be a field, and fix two subgroups ν ⊆ ν ′ of F∗. (Typically, we have in mindν = {1} or {±1}, and ν ′ the torsion subgroup of the units of the ring of algebraic integers in anumber field.) Let ∆ = GL2(F)/ν. Let L be the set of orbits for the action of ν ′ on F2 \{(0, 0)}given by scalar multiplication, which has a natural map to P1

F . The extreme cases ν ′ = F∗and ν ′ = {1} give L = P1

F and L = F2 \ {(0, 0)} respectively. For n ≥ 0 we let Cn(L) be thefree abelian group with as generators (n+ 1)-tuples (l0, . . . , ln) of elements in L such that if li1and li2 have the same image in P1

F then li1 = li2 (see Remark 3.15 for an explanation of thiscondition). We shall call such a tuple (l0, . . . , ln) with all li distinct in L (or equivalently, in P1

F )non-degenerate, and we shall call it degenerate otherwise. Then ∆ acts on Cn(L) as ν ⊆ ν ′, andwith the usual boundary map d:Cn(L)→ Cn−1(L) for n ≥ 1 given by

d((l0, . . . , ln)) =n∑i=0

(−1)i(l0, . . . , li, . . . , ln),

where li indicates that the term li is omitted, we get a complex

(3.11) · · · C4(L) C3(L) C2(L) C1(L) C0(L)d d d d d

of Z[∆]-modules.For three non-zero points p0, p1 and p2 in F2 with distinct images in P1

F , we define cr2(p0, p1, p2)

in ∧2F∗ by the rules:

• cr2(gp1, gp2, gp3) = cr2(p1, p2, p3) for every g in GL2(F);• cr2((1, 0), (0, 1), (a, b)) = (a)∧(b). 1

Because cr2((0, 1), (1, 0), (a, b)) = (b)∧(a) and cr2((1, 0), (a, b), (0, 1)) = (−ab−1)∧(b−1) = −(a)∧(b),we see that cr2 is alternating. It is also clear that if we scale one of the pi by λ in ν ′ thencr2(p0, p1, p2) changes by a term (λ)∧(c) with c in F∗. Let

(3.12) ∧2F∗/ν ′∧F∗ =∧2F∗

〈(λ)∧(c) with λ in ν ′ and c in F∗〉.

We then obtain a group homomorphism

f2,F :C2(L)→ ∧2F∗/ν ′∧F∗

by letting this be trivial on a degenerate generator (l0, l1, l2), and by mapping a non-degenerategenerator (l0, l1, l2) to cr2(p0, p1, p2) with pi a point in li. (Clearly ν ′ should be part of thenotation, but we suppress it for this map.)

1Goncharov in [23, §3] maps this to (−1)∧(−1) + (b)∧(a).


We next define a group homomorphism

f3,F :C3(L)→ Z[F []

as follows. On a degenerate generator (l0, l1, l2, l3) we let f3,F be trivial, and we let it map a

non-degenerate generator (l0, l1, l2, l3) to [cr3(l0, l1, l2, l3)], the generator corresponding to thecross-ratio cr3 of the images of the points in P1

F . Recall that cr3 is defined by rules similar tothose for cr2:

• cr3(gl0, gl1, gl2, gl3) = cr3(l0, l1, l2, l3) for every g in GL2(F);

• cr3([1, 0], [0, 1], [1, 1], [x, 1]) = x for x in F [.

Remark 3.13. From the GL2(F)-equivariance of cr3 one sees by a direct calculation that, forl0, l1, l2, l3 different non-zero points in F2,

cr3(l0, l1, l2, l3) =det(

[l1 l3

]) det(

[l2 l4

])

det([l1 l4

]) det(

[l2 l3

]).

As is well known, from this, or by a direct calculation, we see that permuting the four pointsmay result in the following related possibilities for a cross ratio: x, 1− x−1, (1− x)−1 for evenpermutations, and 1 − x, x−1, (1 − x−1)−1 for odd ones, with the subgroup V4 of S4 actingtrivially.

3.3.2. In the next result we consider the homomorphism

δν′

2,F :Z[F []→ ∧2F∗/ν ′∧F∗

that sends each element [x] for x in F [ to the class of (1− x)∧x. If ν ′ is trivial then this is stillthe map δ2,F of (3.2).

Lemma 3.14. The following diagram commutes.

C3(L) C2(L)

Z[F [] ∧2F∗/ν ′∧F∗

d

f3,F f2,F

δν′

2,F

Proof. It suffices to check commutativity for each generating element (l0, l1, l2, l3) of C3(L.)If (l0, l1, l2, l3) is non-degenerate, this follows by an explicit computation. Specifically, by using

the GL2(F)-invariance of both f3,F and f2,F and the GL2(F)-equivariance of d one can assumethat (l0, l1, l2, l3) are the classes of (a, 0), (0, b), (1, 1) and (xc, c) in L for some and a, b and c

in F∗ and x in F [, which results in [x] in Z[F [] under f3,F and the class of (1 − x)∧(x) underf2,F ◦ d.

For a degenerate tuple (l0, l1, l2, l3) the commutativity is obvious if {l0, l1, l2, l3} has at mosttwo elements as then f2,F is trivial on every term in d((l0, l1, l2, l3)).

If {l0, l1, l2, l3} consists of three classes A, B and C with A occurring twice among l0, l1,l2 and l3, then up to permuting B and C the six possibilities for (l0, l1, l2, l3) are (A,A,B,C),(A,B,A,C), (A,B,C,A), (B,A,A,C), (B,A,C,A) and (B,C,A,A). After cancellation of iden-tical terms with opposite signs in d((l0, l1, l2, l3)), we see that commutativity follows because f2,Fis alternating. �


Remark 3.15. The argument used to prove Lemma 3.14 provides the motivation for consideringonly tuples (l0, . . . , ln) of elements in L such that if li1 and li2 have the same image in P1

F thenli1 = li2 . It seems reasonable to define f2,F and f3,F to be trivial on tuples for which some pointshave the same image in P1

F . Starting with such a tuple (A,A′, B,C) where A and A′ have thesame image, but A, B and C have different images, we require f2,F takes the same value on(A,B,C) and (A′, B,C) and so must limit the amount of scaling between A and A′ to ν ′.

3.3.3. We now set

p(F) =Z[F []

(f3,F ◦ d)(C4(L)).

Then the diagram in Lemma 3.14 induces a commutative diagram

(3.16)

· · · C4(L) C3(L) C2(L) C1(L) C0(L)

0 p(F) ∧2F∗/ν ′∧F∗ 0

d d

f3,F

d

f2,F

d d

∂ν′

2,F

in which ∂ν′

2,F denotes the map induced by δν′

2,F . (If ν ′ is trivial then we shall use the notation

∂2,L for the induced map.) We observe that we could take GL2(F)-coinvariants in the toprow because of the properties of f3,F and f2,F . In particular, f3,F induces a homomorphism

H3(C•(L)GL2(F))→ B(F)ν′ , where we let

B(F)ν′ = ker(∂ν′

2,F ) .

We shall denote this simply B(F) if ν ′ is trivial.

3.3.4. We next show that the group B(F) defined above is naturally isomorphic to a quotientof the ‘Bloch group’ B(F) that is defined by Suslin in [42].

This result motivates us to regard B(F) as a modified Bloch group (and therefore explainsour choice of notation). Before proving it, we first recall the group defined in [42].

For an infinite field F , Suslin considers the group

p(F ) =Z[F []

〈[x]− [y] + [ yx ] + [1−x1−y ]− [1−x−1

1−y−1 ] with x, y in F [, x 6= y〉.

He then defines B(F) to be the kernel of the homomorphism

p(F)→ (F∗ ⊗F∗)σ

[x] 7→ xσ⊗ (1− x) ,

where we set

(F∗ ⊗F∗)σ :=F∗ ⊗F∗

〈x⊗ y + y ⊗ x with x, y in F∗〉

and write aσ⊗ b for the class in the quotient of an element a⊗ b.

We further recall from loc. cit. that Suslin’s proves the existence of a canonical short exactsequence of the form

(3.17) 0→ Tor(F∗,F∗)∼ → K3(F)ind → B(F)→ 0,


where Tor(F∗,F∗)∼ denotes the unique non-trivial extension of Tor(F∗,F∗) by Z/2Z if F hascharacteristic different from 2, and denotes Tor(F∗,F∗) otherwise, and K3(F)ind is the cokernelof the natural homomorphism from the Milnor K-group KM

3 (F) to K3(F).

We also recall that the element cF = [x] + [1− x] of B(F) is independent of x in F [ and hasorder dividing 6 [42, Lem. 1.3, 1.5] and that the order of cQ in B(Q) is equal to 6 [42, Prop. 1.1].

3.3.5. The group p(F) that we defined above is obtained from p(F) by quotienting out by the

subgroup that is generated by all elements of the form [x] + [x−1] with x in F [ and [y] + [1− y]

with y in F [.Since the latter elements generate the same group as does the element cF defined above we are

motivated to consider the following short exact sequence of complexes (with vertical differentials)

0 〈[x] + [x−1] with x in F [〉 p(F)/〈cF 〉 p(F) 0

0 〈xσ⊗ (−x) with x in F [〉 (F∗ ⊗F∗)σ ∧2F∗ 0.

f ∂2,F

Theorem 3.18. If F is infinite, then the homomorphism f in the above diagram is bijective.In particular, in any such case the diagram induces an isomorphism B(F)/〈cF 〉 → B(F).

Proof. A direct calculation shows that f maps the class of [x] + [x−1] to xσ⊗ (−x), so f is

surjective.In order to show that f is injective, recall that by [42, Lem. 1.2], the map F∗ → p(F) sending

x to [x] + [x−1] if x 6= 1 and 1 to 0, is a homomorphism, and that (F∗)2 is in its kernel. We shallconsider its composition with the quotient map to p/〈cF 〉, giving a surjective homomorphism

g:F∗ → 〈[x] + [x−1] with x in F [〉 ,with the target in p(F)/〈cF 〉. If −1 is a square then we already know that [−1] + [−1] = 0in p(F). If −1 is not a square then 2 6= 0, so 2[−1] = 2cF − 2[2] = 2cF + 2[1

2 ] = 3cF in p(F)

(cf. [42, Lem. 1.4]). In either case, we have that {±1} · (F∗)2 ⊆ ker(g), and that im(g) is the

subgroup generated by the classes of [x]+[x−1] with x in F [. We also want to consider ker(f ◦g).For this, we fix a basis B of F∗/(F∗)2 as F2-vector space, making sure to include −1 in B if−1 is not a square in F∗. For b in B, the homomorphism F∗/(F∗)2 → F2 · b ' F2 obtainedfrom the projection onto F2 · b can be applied twice in the tensor product in order to give acomposite homomorphism F∗ ⊗ F∗ → F∗/(F∗)2 ⊗ F∗/(F∗)2 → F2 ⊗ F2 ' F2. This induces a

homomorphism hb: (F∗ ⊗ F∗)σ → F2, mapping xσ⊗ y to the product of the coefficients of b in

the classes of x and y in F∗/(F∗)2. If x in F∗ is in ker(f ◦ g), then hb(xσ⊗ (−x)) = 0 for all b. If

−1 is a square, this means x is a square. If −1 is not a square, then x or −x must be a square.In either case, it follows that ker(f ◦ g) ⊆ {±1} · (F∗)2. Because {±1} · (F∗)2 ⊆ ker(g) and g issurjective, it follows that f is injective.

So f is an isomorphism. The isomorphism B(F)/〈cF 〉 → B(F) now follows from the snakelemma. �

Remark 3.19. Note that in the proof above, it also follows that ker(g) = {±1}(F∗)2. So ginduces an isomorphism from F∗/{±1} · (F∗)2 to the subgroup of p(F)/〈cF 〉 generated by the


classes of [x]+[x−1] with x in F [, given by mapping the class of x to the class of [x]+[x−1]. Andf ◦ g induces an isomorphism from F∗/{±1} · (F∗)2 to the subgroup of (F∗ ⊗ F∗)σ generated

by the xσ⊗ (−x), mapping the class of x to the class of x

σ⊗ (−x).

3.3.6. We can now state the main result of this section. This result concerns the map ϕF inTheorem 3.4.

Theorem 3.20. ϕF induces a homomorphism ψF :B(F) → K3(F)indtf .

Proof. We claim first that the subgroup (f3,F ◦ d)(C4(L)) of Z[F [] is generated by all elementsof the form

(3.21) [x]− [y] + [y/x]− [(1− y)/(1− x)] + [(1− y−1)/(1− x−1)] .

for x 6= y in F [, and

(3.22) [x] + [x−1] and [y] + [1− y]

for x and y in F [.To see this we note that for each non-degenerate generator (l0, . . . , l4) of C4(L) one has

(f3,F ◦ d)((l0, . . . , l4)) =5∑i=1

(−1)i cr3(l0, . . . , li, . . . , l4),

where l0, . . . , l4 are distinct points in P1F and li indicates that the term li is omitted.

In view of the invariance of cr3 under the action of GL2(F), and the fact that for cr3 we canuse points in P1

F , we may assume the points are (1, 0), (0, 1), (1, 1), (x, 1) and (y, 1) for x 6= y in

F [. This then under f3,F ◦ d yields the element (3.21).Let now (l0, . . . , l4) be a degenerate generator. Then its image under f3,F ◦ d is trivial if

{l0, . . . , l4} has at most three elements since then all the terms in d((l0, . . . , l4)) are degenerate.On the other hand, if {l0, . . . , l4} has four elements, then after cancelling possible identical termsin d((l0, . . . , l4)) and applying f3,F to the result we see that it is of the form

[cr3(m1, . . . ,m4)]− sgn(σ)[cr3(mσ(1), . . . ,mσ(4))]

for a permutation σ in S4 with sign sgn(σ), and four distinct points mi in P1F . The subgroup

generated by these images coincides with the subgroup generated by the terms (3.22). (Thisshows, in particular, that the map f3,F is alternating.)

This justifies the above claim and so to prove the proposition, it is enough to show that ϕFis trivial on all elements of the form (3.21) and (3.22).

If F is a number field F then this follows from Theorem 3.4(iii), (3.3), and Borel’s theorem,Theorem 2.1, by letting σ run through all embeddings of F into C.

In order to see that it holds for all fields F as in Theorem 3.4, we can tensor with Q, in whichcase the construction underlying the construction of the map ϕF in Theorem 3.4 is the simplestcase of the constructions that are made by the second author in [16].

One can then verify that the elements in (3.22) and (3.21) are trivial by working overZ[x, x−1, (1−x)−1] or Z[x, y, (1−x)−1, (1− y)−1, (x− y)−1] as the base schemes, along the linesof the proof of [16, Prop. 6.1] where the base scheme can be taken to be Z[x, x−1, (1−x)−1]. Weleave the precise details of this argument to an interested reader. �


Remark 3.23. If ν ′ is finite of order a, then multiplying an element in B(F)ν′ in the bottomrow of (3.16) with a gives an element in B(F).

Remark 3.24. The map Z[k[] → R(1) in Theorem 3.4(iii), mapping [x] to iD(σ(x)) for anembedding σ: k → C, induces a map Dσ: p(k)→ R(1) because the relations in (3.21) and (3.22)are matched by (3.3).

Remark 3.25. In this remark we assume ν ′ to be trivial and explain the advantages to ourapproach of the definitions that we have adopted in comparison to those that are used byGoncharov in [23].

To be specific, we recall that in (3.8) of loc. cit., a key role is played by the boundary map

(3.26) Z[F []→ F∗ ⊗F∗

〈a⊗ b+ b⊗ a with a, b in F∗〉

that sends a generator [x] to the class of x⊗ (1−x). Our group ∧2F∗ is a quotient of the targetgroup here (cf. the diagram just before Theorem 3.18), and δ2,F maps [x] to the image of the

class of x⊗ (1− x) in ∧2F∗.Now the map that Goncharov constructs from non-degenerate triples of non-zero points in F2

to the right hand side of (3.26) is not itself GL2(F)-equivariant since letting a matrix withdeterminant c act changes the result by the class of c⊗ (−c).

In addition, the calculation with the four points (a, 0), (0, b), (1, 1) and (xc, c) in the proof ofLemma 3.14 would similarly result in the class in the right hand side of (3.26) of the elementx⊗ (1− x) + c⊗ (−c) which is not what one wants.

Whilst these problems could be simply resolved by multiplying any of the relevant maps by afactor of two, this would in the end lead either to a smaller subgroup of K3(F)ind

tf if we multiplyf3,F by 2, or a (new) Bloch group that is too large (if we multiply Goncharov’s boundary mapby 2). It is therefore better for us to avoid the problem by replacing the right hand side of (3.26)

as the target of the boundary map by its quotient ∧2F∗.On the other hand, the elements of the form [x] + [x−1] that are in the kernel of δ2,F could

then result in a potentially large and undesired subgroup in the kernel of the boundary map,even modulo the 5-term relations (3.21) (see Theorem 3.18 and its proof). To avoid this possibleproblem we have also imposed the relations (3.22) when defining p(k) by working with degenerateconfigurations.

3.4. Torsion elements in Bloch groups. In this section we study the torsion subgroup ofthe modified Bloch group B(F ) of a number field F by means of a comparison with the Blochgroup B(F ) defined by Suslin (and recalled in §3.3.4).

In this way, we find that B(F ) is torsion free if F is equal to either Q, or to an imaginaryquadratic number field or is generated over Q by a root of unity (of any given order). In thecase of imaginary quadratic fields, this fact will then play an important role in § 4.

The computations made here will also establish the precise relation for any infinite field Fbetween our groups B(F) and p(F) and the corresponding groups B(F) and p(F) recalled in§3.3.4.

3.4.1. For the sake of simplicity, we formulate and prove the next result only for number fields.


In its statement, if p is a prime number, then we denote the p-primary torsion subgroup ofa finitely generated abelian group by means of the subscript p. Because all torsion groups hereare finite and cyclic, this determines their structures.

Proposition 3.27. Let F be a number field. For a prime p, let ps be the number of p-powerroots of unity in F , and let r be the largest integer such that Q(µpr)

+ ⊆ F . Then the size of thep-power torsion in the various groups are as follows. (Note r ≥ 2 and s ≥ 1 if p = 2, and r ≥ 1if p = 3.)

prime |Tor(F ∗, F ∗)∼p | |K3(F )indp | |B(F )p| |B(F )p| condition

p ≥ 5 ps prpr pr ζp /∈ F1 1 ζp ∈ F

p = 3 3s 3r3r 3r−1 ζ3 /∈ F1 1 ζ3 ∈ F

p = 2 2s+1 2r+1 2r−1 2r−2 ζ4 /∈ F1 1 ζ4 ∈ F

Proof. We compute |K3(F )indp | (which is faster than using [49, Chap. IV, Prop. 2.2 and 2.3]).

Let A ⊆ Z∗p be the image of Gal(Q/F ) in Gal(Q(µp∞)/Q) ' Z∗p. Then there are identifications

K3(F )indp ' H0(Gal(Q/F ),Qp(2)/Zp(2)) ' ∩a∈A ker(Qp/Zp

a2−1→ Qp/Zp),

where the first follows from [34, Cor. 4.6] and the second is clear.We assume for the moment that p 6= 2. Then r = 0 is equivalent with A 6⊆ {±1} · (1 + pZp),

so some a2 − 1 is in Z∗p and this kernel is trivial. For r ≥ 1, we have A ⊆ {±1} · (1 + prZp) but

A 6⊆ {±1} · (1 + pr+1Zp). Then 1 + pZp ⊆ A because p is odd, hence A2 = 1 + prZp and thestatement is clear.

To deal with the case p = 2 we note that A is the image of Gal(Q/F ) in Gal(Q(µ2∞)/Q) 'Z∗2 = {±1} · (1 + 4Z2). Then A ⊆ {±1} · (1 + 2rZ2) but A 6⊆ {±1} · (1 + 2r+1Z2), for some r ≥ 2.In this case, A contains an element of {±1} · (1 + 2rZ∗2), and A2 = 1 + 2r+1Z2, from which thestatement follows.

We always have r ≥ s. If ζ2p is in F then r = s because Q(ζpr)+(ζ2p) = Q(ζpr). If ζ2p is not

in F , then s = 0 for p 6= 2, and s = 1 for p = 2. The entries for |B(F )p| are now immediatefrom (3.17). From this, we recover that B(Q) has order 6, so is generated by cQ. Then we cancompare the sequences (3.17) for the field Q and for F . Using that K3(F )ind

tor is cyclic, and thatcQ maps to cF under the injection K3(Q)ind

tor → K3(F )indtor , it follows that 3cF has order 2 if and

only if ζ4 /∈ F , and that 2cF has order 3 if and only if ζ3 /∈ F : in order to have those orders, wemust have |Tor(F ∗, F ∗)∼p |= |Tor(Q∗,Q∗)∼p | for p = 2 or p = 3 respectively. (Cf. [42, Lem. 1.5].)

This gives the entries for |B(F )p|= |(B(F )/〈cF 〉)p|. �

If F is a number field, and p a prime number, then combining Theorem 3.18 and Proposi-tion 3.27 gives a precise statement when B(F ) has no p-torsion.

Theorem 3.28. Let F be a number field. Then, for a given prime number p, the group B(F )has no p-torsion if and only if the following condition is satisfied:


- if p ≥ 5, then either Q(ζp)+ 6⊆ F or Q(ζp) ⊆ F ;

- if p = 3, then either Q(ζ9)+ 6⊆ F or Q(ζ3) ⊆ F ;- if p = 2, then either Q(

√2) = Q(ζ8)+ 6⊆ F or Q(ζ4) ⊆ F .

Proof. We first state when B(F ) has non-trivial p-torsion, as this is immediate, using Theo-rem 3.18: with notation as in Proposition 3.27, this is the case if and only if

- r ≥ 1 and ζp is not in F , when p ≥ 5;- r ≥ 2 and ζ3 is not in F , when p = 3;- r ≥ 3 and ζ4 is not in F when p = 2.

Upon negating this statement for each prime p one obtains the claimed result. �

Corollary 3.29. We have that

(i) B(Q) is trivial;

(ii) B(F ) ' Z[F :Q]/2 if F = Q(ζN ) with N ≥ 3;(iii) B(F ) ' Z if F is an imaginary quadratic field.

In addition, for each of these fields the composition of the maps ψF :B(F ) → K3(F )indtf and

K3(F )indtf →

∏σ R(1), where σ runs through the places of F , is injective.

Proof. We first let F be any number field. Then B(F ) is a finitely generated abelian groupof the same rank as K3(F ) by (3.17) and Theorem 3.18, hence, by Theorem 3.4(ii), the kernelof ψF is the torsion subgroup of B(F ). Because of the behaviour of the regulator with respectto complex conjugation, in Theorem 2.1(iii) we only have to consider all places of F , not allembeddings into C.

It therefore suffices to check that for the number fields listed in the corollary, B(F ) has nop-torsion for every prime number p. But this follows from Theorem 3.28. �

3.4.2. We conclude this subsection with some statements on the torsion in p(F ) and its be-haviour under field extensions, but for simplicity, we do this only for F = Q or imaginaryquadratic.

Proposition 3.30.

(i) The torsion subgroup of p(Q) has order 2 and is generated by [2].(ii) If k is an imaginary quadratic number field, then the natural map p(Q) → p(k) and its

composition with ∂2,k are injective when k 6= Q(√−1), but for k = Q(

√−1) both kernels

are generated by [2].

Proof. To prove claim (i) we note Corollary 3.29 implies that p(Q) injects into ∧2Q∗. We also

know from Proposition 3.5 (or Corollary 3.8) that the natural map 〈−1〉 ⊗Q∗ → ∧2Q∗ gives anisomorphism with the torsion subgroup of the latter.

So we want to compute the kernel of the natural map 〈−1〉 ⊗ Q∗ → K2(Q). Using the tamesymbol, and the fact that {−1,−1} is non-trivial in K2(Q), one sees that this kernel is cyclic oforder 2, generated by (−1)∧2 = ∂2,Q([2]). And 0 = [1

2 ] + [1− 12 ] = 2[1

2 ] = −2[2].

Turning to claim (ii) we note that, under ∂2,Q, the kernel of the composition must inject into

the kernel of ∧2Q∗ → ∧2k∗, which we computed in Lemma 3.10.


In particular, as this kernel is a torsion group, we see from Corollary 3.29 that the kernel

of the composition and that of p(Q) → p(k) coincide as the torsion of p(k) injects into ∧2k∗

under ∂2,k.

In addition, by claim (i), those kernels are either trivial or generated by [2]. They contain [2]

if and only if (−1)∧2 is in the kernel of ∧2Q∗ → ∧2k∗, which by Lemma 3.10 holds if and only

if k = Q(√−1).

This completes the proof. �

Remark 3.31. Note that [2] = 0 in p(Q(√−1)) follows explicitly from (3.21) with x =

√−1,

y = −√−1, which gives [−1] = 0, as [2] = −[−1] by (3.22).

3.5. A conjectural link between the groups of Bloch and Suslin. If F is an infinite field,

then (3.17) gives an isomorphism K3(F)indtf

'→ B(F)tf and Theorem 3.18 gives an isomorphism

B(F)tf'→ B(F).

By Theorem 3.20, one also knows that the homomorphism ϕF in Theorem 3.4 induces ahomomorphism of the form ψF :B(F) → K3(F)ind

tf . This in turn induces a homomorphism

ψF ,tf :B(F)tf → K3(F)indtf ,

thereby allowing us to form the composition

(3.32) K3(F)indtf

'→ B(F)tf'→ B(F)tf

ψF,tf−→ im(ψF ,tf) ⊆ K3(F)indtf .

The group im(ψF ,tf) that occurs here is essentially the same, modulo torsion, as the groupthat was originally defined by Bloch in [4] and which inspired Suslin to define and study thegroup B(F) in [42]. Both of these groups are described as the kernel of a map from a group that

is generated by elements of the form [x] for x in F [, sending each [x] to the class of (1− x)⊗ xin either F∗ ⊗F∗ or a variant like ∧2F∗.

However, as Bloch’s constriction uses relative K-theory whilst Suslin’s uses group homologyof GL2(F), there is a priori no obvious relation between the groups, nor any obvious way toconstruct a map between them.

It seems, nevertheless, to be widely expected that these groups should be closely related andour approach now provides the first concrete evidence (in situations in which the groups arenon-trivial) to suggest both that these groups should be related in a very natural way, andalso that Bloch’s group should account for all of the indecomposable K3-group, at least modulotorsion.

To be specific, if F is equal to a number field F , then ψF,tf is injective by the proof ofTheorem 3.28, so by Proposition 3.4 the composite map (3.32) is an injection of a finitelygenerated free abelian group into itself. One can therefore try to determine the size of itscokernel (or, equivalently, of the cokernel of ψF,tf) by comparing the result of the regulators on

im(ψF,tf) and on K3(F )indtf .

This observation combines with extensive evidence that we have obtained by computer cal-culations in the case that F is imaginary quadratic (cf. §6) to motivate us to formulate thefollowing conjecture.

Conjecture 3.33. If F is a number field, then ψF,tf is an isomorphism.


Remark 3.34. As mentioned above, the map ψF,tf is injective and so the main point of Conjec-

ture 3.33 is that im(ψF,tf) = K3(F )indtf , and hence that the group defined by Bloch in [4] accounts

for all of K3(F )indtf . However, for a general number field F , this equality would not itself resolve

the problem of finding an explicit description of the resulting composite isomorphism in (3.32).Of course, if F is an imaginary quadratic number field, then all groups occurring in the compos-ite are isomorphic to Z and so the conjecture would imply that (3.32) is multiplication by ±1.(Recall that ψF,tf is itself natural up to a universal choice of sign since this is true for ϕF .)

It would also seem reasonable to hope that (3.32) has a very simple description for any infinitefield F , such as, perhaps, being given by multiplication by some integer that is independent of F .Assuming this to be the case, our numerical calculations would imply that this integer is ±1. Iftrue, this would in turn imply that the isomorphism K3(F)ind

tf → B(F)tf constructed by Suslinin [42] could be given a more direct, and more directly K-theoretical, description, at least up tosign, as the inverse of the composite isomorphism B(F)tf → B(F) → K3(F)ind

tf where the firstmap is induced by Theorem 3.18 and the second is ψF,tf .

4. A geometric construction of elements in the modified Bloch group

Let k be an imaginary quadratic number field with ring of algebraic integers O.In this section, we shall use a geometric construction, the Voronoi theory of Hermitian forms,

in order to construct a non-trivial element βgeo in B(k) ' Z.To do this we shall invoke a tessellation of hyperbolic 3-space for k, based on perfect forms, in

order to construct an element of the kernel of the homomorphism d:C3(L)→ C2(L) that occursin (3.11) with F = k and ν ′ = {1}.

By applying f3,k to this element we shall then obtain βgeo by using the commutativity of thediagram (3.16).

Furthermore, we are able to explicitly determine the image of this element under the regulatormap and compare it to the special value ζ ′k(−1) by using a celebrated formula of Humbert.

This will in particular show that the element of K3 that is constructed in this geometricfashion is of index |K2(O)| in the full group K3(k) (cf. Corollary 4.11(i)).

4.1. Voronoi theory of Hermitian forms. Our main tool is the polyhedral reduction theoryfor GL2(O) developed by Ash [1, Chap. II] and Koecher [30], generalizing work of Voronoi [44]on polyhedral reduction domains arising from the theory of perfect forms. See [50, §3] as wellas [17, §2 and §6] for details and a description of the algorithms involved. We recall some of thedetails here to set notation.

We fix a complex embedding k ↪→ C and identify k with its image. We extend this iden-tification to vectors and matrices as well. We use · to denote complex conjugation on C, thenon-trivial Galois automorphism on k. Let V = H2(C) be the 4-dimensional real vector space of2×2 complex Hermitian matrices with complex coefficients. Let C ⊂ V denote the codimension0 open cone of positive definite matrices. Using the chosen complex embedding of k, we canview H2(k), the 2× 2 Hermitian matrices with coefficients in k, as a subset of V . Define a mapq:O2 \ {0} → H2(k) by q(x) = xxt. For each x ∈ O2, we have that q(x) is on the boundary ofC. Let C∗ denote the union of C and the image of q.

The group GL2(C) acts on V by g · A = gAgt. The image of C in the quotient of V bypositive homotheties can be identified with hyperbolic 3-space H. The image of q in this quotientis identified with P1

k, the set of cusps. The action induces an action of GL2(k) on H and the


cusps of H that is compatible with other models of H (see [19, Chap. 1] for descriptions of othermodels). We let H∗ = H ∪ P1

k.Each A ∈ V defines a Hermitian form A[x] = xtAx, for x ∈ C2. Using the chosen complex

embedding of k, we can view O2 as a subset of C2.

Definition 4.1. For A ∈ C, we define the minimum of A as

min(A) := infx∈O2\{0}

A[x].

Note that min(A) > 0 since A is positive definite. A vector v ∈ O2 is called a minimal vector ofA if A[v] = min(A). We let Min(A) denote the set of minimal vectors of A.

These notions depend on the fixed choice of the imaginary quadratic field k. Since O2 isdiscrete in the topology of C2, a compact set {z | A[z] ≤ bound} in C2 gives a finite set in O2.Thus Min(A) is finite.

Definition 4.2. We say a Hermitian form A ∈ C is a perfect Hermitian form over k if

spanR{q(v) | v ∈ Min(A)} = V.

By a polyhedral cone in V we mean a subset σ of the form

σ =

{n∑i=1

λiq(vi) | λi ≥ 0

},

where v1, . . . , vn are non-zero vectors in O2. A set of polyhedral cones S forms a fan if thefollowing two conditions hold. Note that a face here can be of codimension higher than 1.

(1) If σ is in S and τ is a face of σ, then τ is in S.(2) If σ and σ′ are in S, then σ ∩ σ′ is a common face of σ and σ′.

The reduction theory of Koecher [30] applied in this setting gives the following theorem.

Theorem 4.3. There is a fan Σ in V with GL2(O)-action such that the following hold.

(i) There are only finitely many GL2(O)-orbits in Σ.

(ii) Every y ∈ C is contained in the interior of a unique cone in Σ.

(iii) Any cone σ ∈ Σ with non-trivial intersection with C has finite stabiliser in GL2(O).

(iv) The 4-dimensional cones in Σ are in bijection with perfect forms over k.

The bijection in claim (iv) of this result is explicit and allows one to compute the structure of

Σ by using a modification of Voronoi’s algorithm [17, §2, §6]. Specifically, σ is a 4-dimensional

cone in Σ if and only if there exists a perfect Hermitian form A such that

σ =

∑v∈Min(A)

λvq(v)

∣∣∣∣∣∣ λv ≥ 0

.Modulo positive homotheties, the fan Σ descends to a GL2(O)-tessellation of H by ideal poly-topes. The output of the computation described above is a collection of finite sets Σ∗n, n = 1, 2, 3,of representatives of n-dimensional cells in H∗ that meet H. Let Σ∗ = Σ∗1 ∪ Σ∗2 ∪ Σ∗3. The cellsin Σ∗ each have vertices described explicitly by finite sets of vectors in O2.


4.2. Bloch elements from ideal tessellations of hyperbolic space. The collection of 3-dimensional cells Σ∗3 above gives rise, after choosing a triangulation of each, to an element inB(k), as follows.

For the sake of brevity, we shall write Γ for PGL2(O). We fix a set of representatives for the3-dimensional cells,

Σ∗3 = {P1, P2, . . . , Pm}.We first establish a useful interpretation of a classical formula of Humbert in this setting.

Lemma 4.4. Let ΓPi denote the stabiliser in Γ of Pi. Then one has

m∑i=1

1

|ΓPi |vol(Pi) = −π · ζ ′k(−1) .

Proof. One has

(4.5)m∑i=1

1

|ΓPi |vol(Pi) = vol(PGL2(O)\H) =

1

8π2|Dk|

32 ·ζk(2).

Here the first equality is clear and the second is a celebrated result of Humbert (see [7], wherethe formula is given for general number fields).

The claimed formula now follows since an analysis of the functional equation (2.13) showsthat the final term in (4.5) is equal to −π · ζ ′k(−1). �

We next subdivide each polytope Pi into ideal tetrahedra Ti,j with positive volume withoutintroducing any new vertices,

(4.6) Pi = Ti,1 ∪ Ti,2 ∪ . . . ∪ Ti,ni .

Here we assume that the subdivision is such that the faces of the tetrahedra that lie in theinterior of the Pi match. An ideal tetrahedron T with vertices v1, v2, v3, v4 has volume

vol(T ) = D(cr3(v1, v2, v3, v4)).

Here D denotes the Bloch-Wigner dilogarithm defined in §3.1.2 above and cr3 denotes the cross-ratio discussed in §3.3.1 and the ordering of vertices is chosen so that the right hand side ispositive.

To ease the notation, we let ri,j denote a resulting cross-ratio for Ti,j . We note that, whilstthere is some ambiguity in choosing the order the four vertices of Ti,j when defining this cross-ratio, the transformation rules in Remark 3.13 combine with the relations in (3.22) to implythat the induced element [ri,j ] of p(k) is indeed independent of that choice.

By Corollary A.5 we know that |ΓPi | divides 24, so the coefficients in the next theorem areintegers.

We also note that, by Proposition 3.30, the map p(Q)→ p(k) is injective unless k = Q(√−1),

that the map 2 · p(Q)→ p(k) is always injective, and that 2 · p(Q) is torsion-free. Moreover, the

compositions of those maps to p(k) with ∂2,k: p(k)→ ∧2k∗ are injections in those two cases.

We now state the main result of this section (the proof of which will be given in §5 below).

Theorem 4.7. Let k be an imaginary quadratic number field, with the polytopes Pi and cross-ratios [ri,j ] chosen as above. Then the following claims are valid.


(i) There exists a unique element βQ in the image of p(Q) in p(k), such that the element

βgeo = βQ +m∑i=1

24

|ΓPi |

ni∑j=1

[ri,j ]

belongs to B(k). If k 6= Q(√−2) then βQ belongs to the image of 2 · p(Q). In all cases

the element βgeo is independent of the choice of representatives in Σ∗3, and the resultingsubdivision (4.6) into tetrahedra.

(ii) If no stabiliser of an element in Σ∗3 or Σ∗2 has order divisible by 4, then there is a unique

βQ in p(Q), which lies in 2 · p(Q), such that the element

βgeo = βQ +

m∑i=1

12

|ΓPi |

ni∑j=1

[ri,j ]

belongs to B(k). Moreover, one has 2 · βgeo = βgeo and 2 · βQ = βQ.

Remark 4.8. The situation for k equal to either Q(√−1) or Q(

√−2) is more complicated

because the order of the stabiliser of the (in both cases unique) element of Σ∗3 has order 24.For k = Q(

√−2) it can be subdivided in several different ways, resulting in the exception in

Theorem 4.7(i). In fact, the subdivision in this case determines whether βQ either belongs, ordoes not belong, to 2 ·p(Q), and both cases occur; see the argument in §5 below for more details.

Remark 4.9. It is sometimes computationally convenient to avoid explicitly computing the ele-ment βQ in Theorem 4.7(i). In this regard it is useful to note that the injectivity in Corollary 3.29combines with Theorem 3.4(iii) and the equality D(z) = −D(z) in (3.3) to imply that

2 · βgeo =

m∑i=1

24

|ΓPi |

ni∑j=1

([ri,j ]− [ri,j ]).

Remark 4.10.(i) The result of Theorem 4.7(ii) applies to many fields Q(

√−d), the first when ordered by d

being Q(√−15), Q(

√−30), Q(

√−35), Q(

√−39), and Q(

√−42). In the first, third and fifth case

here one has |K2(O)|= 2 and so Theorem 4.7(ii) provides a generator.

(ii) Example 5.3 below shows that one cannot ignore the condition on the stabilisers of theelements of Σ∗2 in Theorem 4.7(ii). Specifically, in this case the stabilisers of the elements of Σ∗3have order 2 or 3, one element of Σ∗2 has stabiliser of order 2, but βgeo generates B(k).

4.3. Regulator maps and K-theory. As we fixed an injection of k into C, by the behaviourof the regulator map reg2 with respect to complex conjugation (see (3.3)), we can computeregulators by considering only the composition

K3(k)→ K3(C)reg2−→ R(1).

By slight abuse of notation, we shall denote this composition by reg2 as well.

Corollary 4.11. Assume the notation and hypotheses of Theorem 4.7. Then the followingclaims are valid.


(i) The element ψk(βgeo) satisfies

reg2(ψk(βgeo))

2πi= −12 · ζ ′k(−1)

and generates a subgroup of the infinite cyclic group K3(k)indtf of index |K2(O)|.

(ii) If no stabiliser of an element in Σ∗3 or Σ∗2 has order divisible by 4, then |K2(O)| is even,

and ψk(βgeo) generates a subgroup of K3(k)indtf of index |K2(O)|/2.

Proof. Before proving claim (i) we note that for each polytope Pi in Σ∗3 one has

(4.12)√−1 · vol(Pi) =

ni∑j=1

D([ri,j ]),

where D is the homomorphism p(k) → R(1) that is defined in Remark 3.24 with respect to afixed embedding k → C.

This is true because vol(Pi) =∑ni

j=1 vol(Ti,j) whilst for each i and j one has√−1 ·vol(Ti,j) =√

−1 ·D(ri,j) = D([ri,j ]).Turning now to the proof of claim (i), we observe that, since the element βQ that occurs in

the definition of βgeo lies in the image of the map p(Q) → p(k), it also lies in the kernel of thecomposite homomorphism reg2 ◦ψk. One therefore computes that

reg2(ψk(βgeo)) =

m∑i=1

24

|ΓPi |

ni∑j=1

reg2(ψk([ri,j ]))

= 24 ·m∑i=1

1

|ΓPi |

ni∑j=1

D([ri,j ])

= 24√−1 ·

m∑i=1

1

|ΓPi |vol(Pi),

= − 24π√−1 · ζ ′k(−1),

where the second equality follows from Theorem 3.4(iii) and Remark 3.24, the third from (4.12)and the last from Lemma 4.4.

The first assertion of claim (i) follows immediately from this displayed equality and, giventhis, the final assertion of claim (i) follows directly from the equality (2.12).

To prove claim (ii) we note that, under the stated conditions, Theorem 4.7(ii) implies βgeo =

2 · βgeo, so this claim follows from the final assertion of claim (i). �

4.4. A cyclotomic description of βgeo. Let k be an imaginary quadratic field of conductor N

and set ζN := e2πi/N . Then we may fix an injection of k into the cyclotomic subfield F = Q(ζN )of C.

The following result shows that the image under the induced map B(k) −→ B(F ) of theelement βgeo constructed in Theorem 4.7(i) has a simple description in terms of elements con-structed directly from roots of unity. (This result is, however, of very limited practical use sinceit is generally much more difficult to compute explicitly in p(F ) rather than in p(k)).

Proposition 4.13. The image of βgeo in B(F ) is equal to N∑

a∈Gal(F/k)[ζaN ].


Proof. At the outset we note that there is a commutative diagram

K3(k)indtf B(k) K3(k)ind

tf R(1)

K3(F )indtf B(F ) K3(F )ind

tf R(1) ,

' ψk reg2

' ψF reg2

where the isomorphisms are obtained from (3.17), as well as Theorems 3.18 and 3.28, and reg2

is the regulator map corresponding to our chosen embeddings of k and F into C.We further recall (from, for example, [41, Prop. 5.13] with X = Y ′ = Spec(F ) and Y =

Spec(k)) that the composition K3(F ) → K3(k) → K3(F ) of the norm and pullback is givenby the trace, and that the same is also true for the induced maps on K3(F )ind

tf = K3(F )tf and

K3(k)indtf = K3(k)tf . By applying this fact to the element of K3(F )ind

tf corresponding to N [ζN ] in

B(F ), we deduce from the left hand square in the above diagram that there exists an elementβcyc in B(k) that maps to N

∑a∈Gal(F/k)[ζ

aN ] in B(F ).

We now identify Gal(F/k) with a subgroup of index 2 of (Z/NZ)∗, which is the kernel of aprimitive character χ : (Z/NZ)∗ → C∗ of order 2, corresponding to k (so χ(−1) = −1). Thenfrom the above diagram and Theorem 3.4(iii), one finds that (2πi)−1 · reg2(γcyc) is equal to

N

4πi

∑a∈Gal(F/k)

∑n≥1

ζnaN − ζ−naN

n2=

N

4πi

∑a∈Gal(F/Q)

∑n≥1

χ(a)ζnaNn2

=N3/2

4πL(Q, χ, 2)

= −12ζ ′k(−1)

as ζk(s) = ζQ(s)L(Q, χ, s), with the Gauß sum∑

a∈Gal(F/Q) χ(a)ζaN = i√N (see [24, §58]).

(Cf. the more general (and involved) calculation of [51, p. 421], or the calculation in the proofof [15, Th. 3.1] with r = −1, ` = 1 and O = Z.)

According to Theorem 4.7, one has (2πi)−1 · reg2(ψk(βgeo)) = −12 · ζ ′k(−1) and so, by the

injectivity of reg2 on K3(k)indtf (cf. Corollary 3.29) one has ψk(βgeo) = ψk(βcyc).

Since ψk and ψF are injective, it follows that βgeo = βcyc, as required. �

5. The proof of Theorem 4.7

Throughout this section we fix an imaginary quadratic field k with ring of integers O, as inthe statement of Theorem 4.7.

5.1. A preliminary result concerning orbits. We start by proving a technical result thatwill play an important role in later arguments.

We set V = k2 \ {(0, 0)} and let Γ denote either SL2(O) or GL2(O).

Lemma 5.1.

(i) For v in V , O∗ acts on the orbit Γv, and the natural map V → P1k induces an injection

of Γv/O∗ into P1k, compatible with the action of Γ.

(ii) For v1 and v2 in V , the images of Γv1/O∗ and Γv2/O∗ are either disjoint or coincide.


Proof. That O∗ acts on Γv is clear if k 6= Q(√−1) or Q(

√−3) as O∗ = {±1} and Γ contains

±id2. For the two remaining cases, O is Euclidean, and based on iterated division with remainderin O it is easy to find g in SL2(O) with gv = ( c0 ) for some c in k∗, so if u is in O∗, then uv is inthe orbit of v because g−1

(u 00 u−1

)g maps v to uv. Alternatively, this follows immediately from

[19, Chap. 7, Lem. 2.1] because if v = (α, β), and u is in O∗, then (α, β) = (uα, uβ).Now assume that g1v = cg2v with c in k∗ and the gi in Γ. Then v is an eigenvector with

eigenvalue c of the element g−12 g1 in Γ, which has determinant ±1. Hence c is in O∗, and g1v and

g2v give the same element in Γv/O∗. The converse is clear, so we do get the claimed injection,and this is clearly compatible with the action of Γ.

For the last part, suppose that g1v1 = cg2v2 for some c in k∗, vi in V , and gi in Γ. We havejust seen that then c is in O, Then Γv1 = cΓv2 and the result is clear. �

Proposition 5.2. If h is the class number of k, then we can find v1, . . . , vh in V such that P1k

is the disjoint union of the images of the Γvi/O∗.In particular, every element in P1

k lifts uniquely to some Γvi/O∗ and this lifting is compatiblewith the action of Γ.

Proof. By [19, Chap. 7, Lem. 2.1] we may identify Γ\V with the set of fractional ideals of O andhence Γ\V/k∗ = Γ\P1

k with the ideal class group of k.Given this, the claimed result follows directly from Lemma 5.1. �

5.2. The proof.

5.2.1. We first establish some convenient notation and conventions.For a 2-cell, or more generally, any flat polytope with vertices v1, . . . , vn in that order along its

boundary, we indicate an orientation by [v1, . . . , vn] up to cyclic rotation. The inverse orientationcorresponds to reversing the order of the vertices. If we want to denote the face with eitherorientation, we write (v1, . . . , vn). In particular, an orientated triangle is the same as a 3-tuple[v1, v2, v3] of vertices up to the action (with sign) of S3. Similarly, an orientated tetrahedron isthe same as a 4-tuple [v1, v2, v3, v4] up to the action (with sign) of S4. Recall that we definedmaps f3,k and f2,k just after (3.12). As mentioned above, the map f3,k is compatible with theaction of S4 by Remark 3.13 and (3.22). By the properties of cr2 mentioned just before (3.12),the map f2,k is also compatible with the action of S3 on orientated triangles if we lift them toelements of C3(L) for some suitable L.

5.2.2. By our discussion before the statement of the theorem, the uniqueness of elements βQand βQ with the stated properties is clear.

It is also clear that for any element βQ in p(k) the explicit sum βgeo belongs to B(k).

In addition, the uniqueness of βQ combines with the explicit expression for βgeo to imply that

2βQ = βQ, hence that 2βgeo = βgeo.The fact that βgeo is independent of the subdivision (4.6), and of the choice of representatives

in Σ∗3 also follows directly from the equality in Corollary 4.11(i) and the injectivity assertions inCorollary 3.29.

To prove Theorem 4.7 it is therefore sufficient to prove the existence of elements βQ and βQin the stated groups such that the sums βgeo and βgeo belong to B(k) and to do this we shalluse the tessellation.


This argument is given in the next subsection. The basic idea is that, for k different fromQ(√−1) and Q(

√−2), the sum 12 · |ΓPi |−1·

∑nij=1[ri,j ] has integer coefficients and belongs to the

kernel of ∂2,k since the faces of the polytopes Pi with those multiplicities can be matched under

the action of Γ, and that f2,k is invariant under the action of GL2(k) and behaves compatiblywith respect to permutations, just as f3,k.

The precise argument is slightly more complicated because the subdivision (4.6) induces tri-angulations of the faces of the Pi which may not correspond, and the faces themselves may haveorientation reversing elements in their stabilisers.

The matching of faces is then just a way of establishing that the explicit sum βgeo lies in thekernel of ∂2,k.

For the special cases k = Q(√−1), k = Q(

√−2), and k = Q(

√−3), we have to compute

more explicitly, either by using a subdivision or the induced triangulations on the faces of thepolytopes.

5.2.3. We note first that each polytope P in the tessellation of H comes with an orientationcorresponding to it having positive volume. For a face (2-cell) F in the tessellation, we fix anorientation that we denote by [F ], and consider the group ⊕FZ[F ], where we identify [F †] with−[F ] if F † denotes F with the opposite orientation.

To any P we associate its boundary ∂P in this group, where each face has the inducedorientation. As the action of Γ on H preserves the orientation, it commutes with this boundarymap.

We now need to do some counting. For a face [F ], we let ΓF denote the stabiliser of the(non-oriented) face F , and Γ+

F the subgroup that preserves the orientation [F ]. We note that

the index of Γ+F in ΓF is either 1 or 2.

Let P and P ′ be the polytopes in the tessellation that have F in their boundaries. If g isin ΓF then gP = P or P ′, and gP = P precisely when g is in Γ+

F . Therefore Γ+F = ΓF ∩ ΓP .

It is convenient to distinguish between the following two cases for the Γ-orbits of F .

• ΓF = Γ+F . If P and P ′ in Σ∗3 are such that their boundaries each contain an element in

the Γ-orbit of [F ], then P and P ′ are in the same Γ-orbit, hence are the same. Thereforethere is exactly one P in Σ∗3 that contains faces in the Γ-orbit of [F ]. If two faces of P arein the Γ-orbit of [F ], then they are transformed into each other already by ΓP . Hencethe number of elements in the Γ-orbit of F in ∂P is [ΓP : ΓF ]. If P ′ is the element inΣ∗3 that has an element in the Γ-orbit of [F ∗] in its boundary (with P = P ′ and P 6= P ′

both possible), then there are [ΓP ′ : ΓF ] = [ΓP ′ : Γ+F ] elements in the Γ-orbit of [F∗] in

the boundary of P ′.• ΓF 6= Γ+

F . Note that in this case [ΓF : Γ+F ] = 2. Here [F ] and [F ∗] are in the same

Γ-orbit and as above one sees that there is only one element P of Σ∗3 that has elementsin this Γ-orbit in its boundary. Any two such elements can be transformed into eachother using elements of ΓP , so there are [ΓP : Γ+

F ] of those in the boundary of P .

5.2.4. In this subsection we prove Theorem 4.7 in the general case that k is neither Q(√−1),

Q(√−2), nor Q(

√−3).


In this case Corollary A.5 implies that the order of each group ΓPi divides 12, and so boththe formal sum of elements in Σ∗3 given by

πP =m∑i=1

12

|ΓPi |[Pi] ,

and the formal sum of of tetrahedra resulting from the subdivision (4.6)

πT =m∑i=1

12

|ΓPi |

ni∑j=1

[Ti,j ] ,

have integral coefficients.We extend the boundary map ∂ to such formal sums, where for πT the boundary is a formal

sum of ideal triangles, contained in the original faces of the polytopes Pi. (Note that thesubdivision (4.6) may introduce ‘internal faces’ inside each polytopes, but by construction theparts of the boundaries of the tetrahedra here cancel exactly. This also holds after lifting allvertices to O2 as there is no group action involved in order to match them. So we may, andshall, ignore those internal faces.)

The subdivision (4.6) induces a triangulation of all the faces of each Pi in Σ∗3. We let [∆F ]denote the induced triangulation. But if F is a face of such a Pi with [F ] 6= [F †], then theinduced triangulations [∆F ] and [∆F † ] (which may come from a different element in Σ∗3) maynot match. Similarly, if gF and F are both faces of Pi with g in ΓPi , then g[∆F ] and [∆gF ] maynot match.

A typical example of non-matching triangulations is that where F is a ‘square’ [v1, v2, v3, v4]that is cut into two triangles using either diagonal, resulting in the triangulations [v1, v2, v4] +[v2, v3, v4] for [F ] and [v1, v2, v3] + [v1, v3, v4] for [F †]. But the boundary of the orientatedtetrahedron [v1, v2, v3, v4] gives exactly the former minus the latter. It is easy to see (e.g., byinduction on the number of vertices of F ) that by using boundaries of such tetrahedra we canchange any triangulation of [F ] into any other with the same orientation. Because the tetrahedra

are contained in the faces, they have no volume, and the cross ratios of the four cusps is in Q[.We refer to them as ‘flat tetrahedra’, and if [∆1

F ] and ∆2F are two triangulations (with the same

orientation) of an orientated face [F ], we shall write

‘[∆1F ] ≡ [∆2

F ] mod. ∂(flat tetrahedra)’

if [∆1F ]− [∆2

F ] is the boundary of a formal sum of such flat tetrahedra.

In particular, if [∆F ] and [∆F † ] are any triangulations of the faces [F ] and [F †] (so withopposite orientation), then [∆F ] + [∆F † ] ≡ 0 mod. ∂(flat tetrahedra).

We extend the boundary map to the free Abelian group ⊕P∈Σ∗3Z[P ] by linearity. For a

given [F ], in ∂(πP ) we find find 12 · |Γ+F |−1 copies of F (up to the action of Γ).

If [F ] 6= [F †] (i.e., if ΓF = Γ+F ) then we have the same number of copies of [F †], and we deal

with [F ] and [F †] together.We now distinguish four cases in terms of the number of factors of 2 in |ΓF | and |Γ+

F |.(1) |Γ+

F | and |ΓF | are odd. Here F † is not in the same Γ-orbit as F , and in ∂(πP ) the

contribution of their Γ-orbits is 12[F ] + 12[F †], modulo the action of Γ. Then for ∂(πT )

they contribute 12[∆F ] + 12[∆†F ] mod. ∂(flat tetrahedra) and modulo the action of Γ.


(2) |Γ+F | is odd and |ΓF |≡ 2 (mod 4). Here F and F † are in the same Γ-orbit, and in the

boundary of πP , up to the action of Γ, we have 12[F ] = 6[F ] + 6[F †]. Then in ∂(πT ) we

obtain 6[∆F ] + 6[∆†F ] mod. ∂(flat tetrahedra) and modulo the action of Γ.

(3) |Γ+F |≡ |ΓF |≡ 2 (mod 4). This is similar to case (1), but now [F ] and [F †] each occur

with coefficient 6 in ∂(πP ). In ∂(πT ) we obtain 6[∆F ] + 6[∆†F ] mod. ∂(flat tetrahedra),and modulo the action of Γ.

(4) |Γ+F |≡ 2 (mod 4) and |ΓF |≡ 0 (mod 4). This is similar to case (2), but now in ∂(πP ) we

find 6[F ] = 3[F ]+3[F †], hence in ∂(πT ) this gives 3[∆F ]+3[∆†F ] mod. ∂(flat tetrahedra)and modulo the action of Γ.

We see that there exists some α, a formal sum of flat tetrahedra, such that πT+α has boundary,up to the action of Γ, a formal sum with terms [t] + [t†] with t an ideal triangle. Lifting all cuspsto L = Γv1/O∗

∐. . .∐

Γvh/O∗ as in Corollary 5.2, and applying f3,k as in (3.16), we see fromthe the Γ-equivariance of f2,k, and the fact that this map is alternating, so kills elements of the

form [t]+ [t†], that∑m

i=1 12 · |ΓPi |−1·∑ni

j=1[ri,j ]+β′ is in the kernel of ∂O∗

2,k , where β′ is the imageof α.

Note also that β′ lies in the image of p(Q) in p(k). Multiplying by 2 and setting βQ := 2β′ wecomplete the proof of Theorem 4.7(i) in this case.

The proof of Theorem 4.7(ii) is obtained in just the same way after starting with∑m

i=1 6 ·|ΓPi |−1·

∑nij=1[ri,j ] (which has integer coefficients under the stated assumptions). In this case

the coefficients in the above cases (1), (2) and (3) are divided by 2, and the situation case (4) isruled out by the assumptions.

5.2.5. We now consider the special fields Q(√−1), Q(

√−2), and Q(

√−3).

In each of these cases either |ΓPi | does not divide 12 or |O∗| is larger than 2. However, onealso knows that Σ∗3 has only one element and its stabiliser has order 12 or 24 and so the result ofTheorem 4.7(ii) does not apply. It is therefore enough to prove Theorem 4.7(i) for these fields.

If k = Q(√−1), then Σ∗3 is an octahedron, with stabiliser isomorphic to S4. Using that an

ideal tetrahedron with positive volume in this octahedron cannot contain two antipodal points,it is easy to see that the subdivision is unique up to the action of the stabiliser. Hence theresulting element under f3,k is well-defined. Computing it explicitly as

∑4j=1[r1,j ] = 4[ω] one

finds that it is in the kernel of ∂2,k as ω2 = −1, so we can simply take βQ = 0.

If k = Q(√−3), then the polytope is a tetrahedron, with stabiliser isomorphic to A4. Com-

puting its image [r1,1] under f3,k explicitly one finds [ω] with ω2 = ω − 1, and ∂2,k([r1,1]) =

ω∧(1 − ω) = (−1)∧(−1) in ∧2k∗, which has order 2 by Remark 3.7). So we can again take

βQ = 0.If k = Q(

√−2), then the polytope is a rectified cube (i.e., a cuboctahedron), with stabiliser

isomorphic to S4, so it has six 4-gons and eight triangles as faces. By the commutativity of (3.16),for ν ′ = {1}, we can compute ∂2,k(

∑j [r1,j ]) by choosing lifts of all vertices involved, and applying

f2,k to each of the lifted triangles (with correct orientation) of the induced triangulation of the

faces of P1. (This provides an alternative approach for Q(√−1) and Q(

√−3) as well.) Note

that any triangulation of the faces occurs for some subdivision: fix a vertex V and use the coneson all the triangles that do not have V as a vertex.


So we must consider all triangulations. Giving the six 4-gons one of the two possible triangula-

tions at random, resulted in (−1)∧(−1) = (−1)∧2 = ∂2,k([2]) in ∧2k∗. The other triangulations

we obtain from this one by triangulating one or more of the 4-gons in the other way. For each4-gon, this adds the image under ∂2,k of the cross-ratio of the flat tetrahedron corresponding to

it. As the 4-gons are all equivalent under Γ, computing this for one suffices. The result is [2]again. So by Proposition 3.30 we find that ∂2,k(

∑j [r1,j ]) equals either 0 or ∂2,k([2]), and both

occur. By Corollary 3.29, we must take βQ = 0 or [2], and by Proposition 3.30 the latter is notin 2p(Q).

This completes the proof of Theorem 4.7.

5.2.6. We make several observations concerning the above argument.

Remark 5.3. Let k not be equal to Q(√−1), Q(

√−2) or Q(

√−3). One can try to find a better

element than βgeo by going through the calculations in the proof of Theorem 4.7 after replacingπP by an element of the form

∑mi=1M · |ΓPi |−1[Pi] for some positive integer M that is divisible

by the orders of the stabilisers ΓPi . If we start with M equal to the least common multiple ofthe orders |ΓPi |, then we may have to replace M with 2M in the proof at two places in order toensure that the resulting element in p(k) belongs to B(k):

(1) in order to ensure that the boundary ∂ of the resulting analogue of πT is trivial up tothe action of Γ, which is not automatic if some Pi has a face with reversible orientationunder Γ and M · |ΓPi |−1 is odd;

(2) in order to ensure that∑m

i=1M · |ΓPi |−1·∑ni

j=1[ri,j ] + β′ is in the kernel of ∂2,k and not

just ∂O∗

2,k , where M results from (1), and β′ (coming from flat tetrahedra) is in p(Q),

which we view as inside p(k) by Proposition 3.30.

Note that in the second part here we use |O∗|= 2, which excludes k = Q(√−1) or Q(

√−3).

For our k, with Nm: k → Q the norm, the Hermitian form (x, y) 7→ Nm(x)+Nm(y)+Nm(x−y)on C2 has minimal vectors {±(1, 0),±(0, 1),±(1, 1)}. By [17, Th. 2.7], this means that thetriangle with vertices 0, 1 and ∞ is a 2-cell of the tessellation. The element

(0 −11 −1

)in Γ of

order 3 stabilises this triangle while preserving its orientation. Therefore the 3-cells that sharethis triangle as faces have stabilisers with orders divisible by 3, and M is divisible by 3.

Also, the elements(−1 0

0 1

)and ( 0 1

1 0 ) have order 2, and generate a subgroup of Γ of order 4.The first has as axis of rotation the 1-cell connecting 0 and ∞, so the axis of rotation of thesecond, which meets this 1-cell, must meet either a 3-cell, or a 2-cell with vertices 0, ∞, andpurely imaginary numbers. In the first case we start with M divisible by 6. In the secondcase, (2) above ensures that M is even since the 2-cell reverses orientation under ( 0 1

1 0 ). Because

in this remark we are also assuming that k 6= Q(√−2), we know from Corollary A.5 that the

greatest common divisor of the orders of the ΓPi divides 12. So this method could lead to anelement βgeo as in Theorem 4.7(i) but with 24 replaced by either 6, 12, or 24.

Remark 5.4. In our calculations, we find the following for all fields k that differ from Q(√−1),

Q(√−2) and Q(

√−3):

• the gcd of the orders of the stabilisers ΓPi is 6 or 12, i.e., 3 does not occur;• the sum

∑mi=1 12 · |ΓPi |−1·

∑nij=1([ri,j ]− [ri,j ]) belongs to B(k), i.e., β′geo is always divisible

by 2 in the most obvious way. (Note that this element must equal βgeo because of theinjectivity of the composition in Corollary 3.29.)


Unfortunately, we have not been able to prove either of these statements in general.

Remark 5.5. With the same notation as before Theorem 4.7, we consider the element β′ =∑mi=1M · |ΓPi |−1·

∑nij=1[ri,j ] of p(k), where M is the greatest common divisor of the orders

|ΓPi |. Then the proof of Theorem 4.7 shows that there exists a positive divisor e of 2|O∗| such

that ∂2,k(eβ′) is in the image of the composition p(Q) → ∧2Q∗ → ∧2

k∗, and it gives a wayof computing this out of the data of the tessellation. But there are some choices involved, forexample, the pairing up of faces of the Pi under the action of Γ, which may result in an e thatis not optimal.

But one can also do this algebraically, by computing β′ and determining a (minimal) positiveinteger e such that e∂2,k(β

′) is in the image of p(Q). For this we can use Corollary 3.8 and

Remark 3.9: if S is a finite set of finite places of k such that ∧2O∗S ⊂ ∧2k∗ contains δ = ∂2,k(β),

and A = Q∗ ∩ O∗S , then one can compute if eδ is in the image of ∧2A or not. If this is the case

then one can compute its preimage in ∧2Q∗ using Lemma 3.10, algorithmically determine if anelement in there yields the trivial element in K2(Q), and if so, express it in terms of ∂2,Q([x])

with x in Q[.Note that a different subdivision (4.6) might a priori give rise to a different β′ and a different e,

but the other choices are irrelevant in this algebraic approach.

5.3. An explicit example. For the reader’s convenience we illustrate the methods of the proofof Theorem 4.7 and of Remark 5.5 in the special case that k = Q(

√−5). In particular, in this

case we find that both methods give the same element.The lifts to O2 (up to scaling by O∗) of the vertices v1, . . . , v8 in the two elements P1 and P2

of Σ∗3 are the columns of the matrix

(5.6)

(ω + 1−2

10

2ω − 1

2ω

01−ω2

1−1−ω + 1ω + 1

).

Both polytopes are triangular prisms, which we write as [a, b, c;A,B,C], where [a, b, c] and[A,B,C] are triangles, with A above a, etc. Such a prism can be subdivided into orientatedtetrahedra as [a,A,B,C]−[a, b, B,C]+[a, b, c, C], which results in the subdivision of its orientatedboundary as

(5.7) [A,B,C] + [a,A,C]− [a, c, C] + [a, b, B]− [a,A,B] + [b, c, C]− [b, B,C]− [a, b, c] .

Here the first and last terms correspond to the triangular faces, and the middle terms are groupedas pairs of triangles in the rectangular faces.

Then P1 = [v3, v5, v4; v1, v2, v6] with ΓP1 of order 2, generated by g1 = (v1v3)(v2v4)(v5v6)in cycle notation on the vertices of P1 (v7 and v8 are mapped elsewhere). It interchanges theorientated faces [v1, v3, v4, v6] and [v3, v1, v2, v5] of P1, and those faces have trivial stabilisers.The orientated face [v2, v6, v4, v5] is mapped to itself by g1 but its stabiliser is non-cyclic oforder 4, with one of the two orientation reversing elements acting as h1 = (v2v5)(v4v6). The twotriangles [v1, v6, v2] and [v3, v5, v4] are interchanged by g1, and both have stabilisers of order 2,with the one for [v1, v6, v2] generated by (v2v6).

We have P2 = [v1, v8, v3; v2, v7, v5] with ΓP2 of order 3, generated by g2 = (v1v3v8)(v2v5v7).The three orientated faces [v2, v7, v8, v1], [v7, v5, v3, v8] and [v5, v2, v1, v3] are all in the same Γ-orbit and have trivial stabilisers. The two triangles [v1, v8, v3] and [v2, v5, v7] are necessarily


non-conjugate (even ignoring orientation) as ΓP2 has order 3, but both have (orientation non-preserving) stabiliser of order 6, which acts as the full permutation group on their vertices.

We now subdivide both P1 and P2 as stated just before (5.7). This gives an element

π′T :=1

2πT = 3[v3, v1, v2, v6]− 3[v3, v5, v2, v6] + 3[v3, v5, v4, v6]

+ 2[v1, v2, v7, v5]− 2[v1, v8, v7, v5] + 2[v1, v8, v3, v5] .

Applying f3,k to this element results in

β′ = 7[13ω + 2

3 ]− 3[−23ω + 5

3 ] + 3[16ω + 5

6 ]− 2[−16ω + 7

6 ]

in p(k).Using (5.7) one can easily compute the boundary ∂π′T . Because

Σ∗2 = {(v1, v3, v4, v6), (v2, v6, v4, v5), (v3, v4, v5), (v1, v8, v3), (v7, v2, v5)} ,under the action of Γ we can move the resulting triangles to the eight triangles that result fromthe elements of Σ∗2, with one of the ‘squares’ giving rise to four inequivalent triangles due to thetwo ways of triangulating a ‘square’, the other to only one inequivalent triangle.

In (⊕tZ[t])Γ, where t runs through the triangles, the triangular faces in ∂(π′T ) coming fromthose of P1 and P2 cancel under the action of Γ (this uses that the coefficient of [P2] in π′T iseven and the triangular faces of P2 have orientation reversing elements in their stabilisers). Ofcourse, the ‘internal’ triangles created by the subdivision into tetrahedra always cancel. Usingg1, g2, h1, h2 one moves the triangles coming from the ‘square’ faces to the five inequivalenttriangles coming from [v1, v3, v4, v6] and [v2, v6, v4, v5]. This yields the sum of the elements

∂[v4, v3, v1, v6] = 3([v3, v1, v6]− [v3, v4, v6] + [v1, v6, v4]− [v1, v3, v4])

+ 2([v3, v4, v6]− [v3, v1, v6] + 2([v1, v3, v4]− [v1, v6, v4]))

and3[v5, v4, v6]− 3[v5, v2, v6] = [v5, v4, v6]− [v5, v2, v6]− ∂[v5, v2, v4, v6] ,

where we used h1.So for πT = 2π′T = 6[P1] + 4[P2] we get ∂(πT − 2∂[v4, v3, v1, v6] + 3∂[v5, v2, v4, v6]) = 0 modulo

the action of Γ. After multiplying by 2 in order to deal with the ambiguity of the lifts of thecusps to O2, we then find the element

βgeo = 4β′ − 4[3] + 6[4/5] ∈ ker(∂O∗

2,k ),

with the last two terms arise because cr3([v4, v3, v1, v6]) = 3 and cr3([v5, v2, v4, v6]) = 4/5.To see if one could do better, as in Remark 5.5, we, instead, compute ∂2,k of β′. This

can be done easily using the matching of triangles under the action of Γ as before, using thecommutativity of (3.16) for ∂O

∗2,k , but as for this we have to lift the vertices to the column vectors

in O2 in (5.6) (and not up to scaling by O∗) we pick up some additional torsion along the way.Alternatively, one can choose a finite set of finite primes S for k such that, for every [z] occurring

in β′, both z and 1 − z are S-units, and compute in ∧2k∗ as in Remark 3.9 and Corollary 3.8.

The result is−(−4)∧(5) + (−2)∧(3) + (−1)∧(2) + (−1)∧(−ω)− 2∧(−5)

in ∧2k∗. Note that [v5, v4, v6] − [v5, v2, v6] under f2,k is mapped to (−1

2)∧(−ω2 ) − 2∧(−ω) =

(−1)∧(−ω) − 2∧(−5). The first three terms are in the image of p(Q), and if we multiply


the last by 2 then we obtain −4∧(−5) = −(−4)∧(−5) = −(−4)∧5 + (−1)∧(−1), with thefirst again in the image of p(Q). If (−1)∧(−1) would come from p(Q) then it would comefrom its torsion by Proposition 3.30(ii), which is generated by [2]. By Lemma 3.10, the kernel

of ∧2Q∗ → ∧2k∗ has order 2, and is generated by (−1)∧(−5), and one easily checks using

Corollary 3.8 that both (−1)∧(−1) and (−1)∧(−1) + (−1)∧(−5) in ∧2Q∗ are neither trivial

nor equal to ∂2,Q([2]) = (−1)∧2. Therefore (−1)∧(−1) in ∧2k∗ is not in the image of p(Q).

Multiplying by 2 again kills the term (−1)∧(−1), hence 4β − 4[3] + 6[5] is in B(k) is the bestpossible for our choice of subdivision. (Note this element equals βgeo above as [4

5 ] = −[15 ] = [5].

Also note that these calculations also show that 2β′ − 2β′ is in B(k), in line with Remark 5.4,and that this element must also equal βgeo.)

In fact, K2(O) is trivial by [2, §7], so by Corollary 4.11(i), γgeo is a generator of the infinite

cyclic group K3(k)ind = K3(k)indtf , βgeo is a generator of the infinite cyclic group B(k), and the

map ψk:B(k) → K3(k)indtf = K3(k)ind is an isomorphism.

This shows that one cannot improve upon the above by using a different triangulation, andthat the obstruction of ‘incompatible lifts’ under the action of Γ is non-trivial, so that both ofthe factors of 2 mentioned in Remark 5.3 are necessary in this case.

6. Finding a generator of K3(k), and computing |K2(Ok)|

In this final section, we again restrict to the case that F is an imaginary quadratic field k andset O := Ok. For simplicity of exposition, we fix an embedding σ : k → C and use it to regardk as a subfield of C.

We explain how to combine an implementation of an algorithm of Tate’s, which produces anatural number that is known to be divisible by the order of K2(O), with either the result ofCorollary 4.11(i) or just the known validity of the precise form of Lichtenbaum’s conjecture fork and m = 2 (cf. Remark 2.11), to deduce our main computational results.

At this stage we have successfully applied the first of these approaches to about twenty fieldsand the second to hundreds of fields.

In this way, for example, we have for all imaginary quadratic fields of discriminant bigger than−1000 determined both the order of K2(O), where not yet known, and a generator of the infinitecyclic group K3(k)ind

tf that lies in the image of the injective homomorphism ψk constructed inTheorem 3.20 (thereby verifying that k validates Conjecture 3.33) and hence also the Beilinsonregulator value R2(k).

The results are available at

https://mathstats.uncg.edu/yasaki/data/

In particular, for each of the listed imaginary quadratic number fields k, the element βalg is such

that its image ψk(βalg) generates K3(k)indtf , thus verifying Conjecture 3.33 for all those fields.

The element βgeo is the element of Theorem 4.7, obtained in the way described in Remark 5.4.

6.1. Dividing βgeo by |K2(Ok)|.

6.1.1. The basic approach is as follows.An implementation by Belabas and Gangl [2] of (a refinement of) an algorithm of Tate gives

an explicit natural number M which is known to be divisible by |K2(O)|.



Since M is typically sharp as an upper bound on |K2(O)|, for any element αgeo of Z[k[] thatlifts βgeo, we try to find an element α in the kernel of δ2,k for which the difference M · α− αgeo

lies in the subgroup generated by the relations (3.21) and (3.22).If one finds such an element α, then its class β in B(k) is such that βgeo = M · β. From the

result of Corollary 4.11(i) it would then follow that |K2(O)|= M , that β generates B(k), thatψk(β) generates K3(k)ind

tf and hence, by Theorem 3.4(iii), that R2(k) = |reg2(ψk(β))|.

6.1.2. To find a candidate element α as above we first use the methods described in §6.3 belowto identify an element α for which one can verify numerically that M ·Dσ(α) = Dσ(αgeo), with Dσthe homomorphism from Remark 3.24.

We then aim to prove algebraically that M · β = βgeo by writing the difference M · α − αgeo

as a sum of explicit relations of the form (3.21) and (3.22).To complete this last step we use a strategy that can be used to investigate whether any

element of the form∑

i ni[xi], where the ni are integers and the xi are in k[, can be written asa sum of such relations.

Here we assume that, for any finite subset U of k[, we work with respect to the Z-basis{[u] with u in U} of Z[U ] ⊂ Z[k[].

We first take U to consist of all elements xi and their images under the 6-fold symmetries thatare used in the relations (3.22), i.e., for u in U we also adjoin 1−u, u−1, 1−u−1, (1−u−1)−1 = −u

1−uand (1− u)−1 to U .

For u 6= v in U , we then consider the element in Z[k[] that is obtained by putting x = u andy = v in (3.21). We use only the resulting relations where all five terms are in U .

We then form a matrix A of width |U |, as follows.

• For the first row we write∑

i ni[xi] in terms of the basis U .• For each of the, n say, 5-term relations that we have just generated, we add a row writing

it in terms of the basis.• For each u in U we add rows corresponding to the relations [u] + [1 − u], [u] + [u−1],

[u]− [1− u−1], [u] + [ −u1−u ] and [u]− [(1− u)−1], resulting in, say, m rows in total.

Then the kernel of the right-multiplication by A on Z1+n+m (regarded as row vectors) givesthe relations among the various elements that we put into the rows of A. In particular, if wefind an element in this kernel that has 1 as its first entry, then we have succeeded in writing∑

i ni[xi] explicitly in terms of the elements of the form (3.21) and (3.22).Unfortunately, however, this rather straightforward method is rarely successful. Instead, we

may have to allow increases in the set U , which may make the computation too large to becarried out. It was, however, done successfully, to some extent by trial and error, for a numberof imaginary quadratic number fields.

Example 6.1. The most notable example among those is the field k = Q(√−303), for which it

is known from [2] that |K2(O)|= 22. The results for this case are described in Appendix B.

Remark 6.2. We note that the method described above for verifying identities in B(k) onlydepends on the definition of B(k) in terms of the boundary map δ2,k and the relations (3.21)

and (3.22) on Z[k[] that are used to define p(k). In particular, it does not rely on knowingthe validity of Lichtenbaum’s Conjecture and so, in principle, the same approach could be usedto show that an element is trivial in p(F ) for any number field F (although, in practice, thecomputations are likely to quickly become unfeasibly large).


6.2. Finding a generator of K3(k)indtf directly. This approach relies on the effective bounds

on |K2(O)| that are discussed above, the known validity of Lichtenbaum’s Conjecture as inRemark 2.11, and an implementation of the ‘exceptional S-unit’ algorithm (see §6.3 below) thatproduces elements in B(k).

In particular, the reliance on Lichtenbaum’s Conjecture means that the general applicabilityof this sort of approach is currently restricted to abelian fields.

To describe the basic idea, we assume to be given an element γ that equals Nγ times a

generator of the (infinite cyclic) group K3(k)indtf for some non-negative integer Nγ .

Then one has |reg2(γ)|= Nγ ·R2(k) and so Remark 2.11 implies that

− reg2(γ)

12 · ζ ′k(−1)=

Nγ

|K2(O)|.

If one now also has an explicit natural number M that is known to be divisible by |K2(O)|,then one knows that the quantity

(6.3) −M reg2(γ)

12 · ζ ′k(−1)= Nγ

M

|K2(O)|

is a product of a non-negative and a positive integer.Hence, if one finds that the left hand side of this equality is numerically close to a natural

number dγ , then Nγ and M/|K2(O)| are both divisors of dγ .In particular, if one can find an element γ for which dγ = 1, then one conclude both that

Nγ = 1 (so that γ is a generator of K3(k)indtf ) and that |K2(O)|= M . We would therefore have

identified a generator of K3(k)indtf and determined |K2(O)|.

To find suitable test elements γ we use the method described in §6.3 below to generateelements α in the subgroup ker(δ2,k) of Z[k[]. We then take γ to be the image under ψk of the

image of α in B(k).Note here that it is not a priori guaranteed that im(ψk) contains a generator of K3(k)ind

tf .However, if this is the case (as it was in all of the examples we tested), then ψk is surjective andso one has verified that k validates Conjecture 3.33.

6.3. Constructing elements in ker(δ2,k) via exceptional S-units.

6.3.1. In order to find enough elements in ker(δ2,k) we fix a set S of finite places of k, andconsider ‘exceptional S-units’, where an exceptional S-unit is an S-unit x such that 1−x is alsoan S-unit.

To compute with such elements it is convenient to fix a basis of the S-units (modulo torsion)of k and then encode exceptional S-units in terms of the exponents that arise when they areexpressed in terms of the fixed basis, and a suitable root of unity. (This is implemented inGP/PARI [43] as ‘bnfsunits’.)

We can then use Corollary 3.8 and Remark 3.9(ii) in order to compute effectively with the

image in ∧2k∗ of the elements (1− x)∧x for the exceptional S-units x.

Example 6.4. In the case k = Q(√−11) and S = {℘2, ℘3, ℘3} where ℘2 = (2) is the unique

prime ideal of norm 4 and ℘3 and ℘3 denote the two prime ideals of norm 3 in O, PARI provides

the S-unit basis B = {b1, b2, b3} with b1 = 2, b2 = −1+√−11

2 and b3 = −1−√−11

2 . We find the


exceptional S-unit x = 536 −

√−1136 of norm 1

36 , for which 1− x has norm 34 , and write

x = −b−11 b−2

2 , 1− x = −b−11 b−2

2 b33 .

It follows that

(1− x)∧x = (−1)∧(−1) + (−1)∧b1 + (−1)∧b3 + 3(b1∧b3) + 6(b2∧b3),

which corresponds to the element (1, 1, 0, 1, 0, 3, 6) under the isomorphism in Corollary 3.8.

This approach effectively reduces the problem of finding elements in ker(δ2,k) to a concreteproblem in linear algebra. Of course, one wants to choose a finite set S of finite places for whichone can find sufficiently many exceptional S-units in k such that some linear combination of themin ker(δ2,k) represents a non-trivial element in (and preferably a generator of) the quotient B(k).

Note that, whilst one can check for non-triviality of an element β in B(k) by simply verifyingthat its image under the map D is numerically non-trivial, in order to conclude that β is trivial,we need to know an explicit natural number M that is divisible by |K2(O)|. Then the quantityon the left hand side of (6.3) is numerically close to zero if and only if γ = ψk(β) is trivial. Ifthat is the case, then the injectivity of ψk implies that the element β is itself trivial.

6.3.2. Since the map ψk has finite cokernel there always exists a finite set S that results in anon-trivial element B(k) by the above procedure.

In practical terms, the geometric approach in §4 shows that one need only take S = Sgeo tobe set comprising all of the places that divide any of the principal ideals O · ri,j and O · (1− ri,j)for the elements ri,j that occur in Theorem 4.7.

In general, this set Sgeo is far too large to be practical for the exceptional S-unit approach.Fortunately, however, in all of the cases investigated in this paper we have found that a muchsmaller choice of set S is sufficient. In fact, we have found that it is often enough to take S tocomprise all places that divide either 2 or 3 or any of the first ten (say) primes that split in k.

Example 6.5. In the case k = Q(√−303) it suffices to take S to be the set of places that divide

either of 2, 11 and 13 (all of which split in k) or 3 (which ramifies in k). Imposing small boundson the exponents with respect to a chosen basis, we already find 683 exceptional S-units in k.Setting ω := (1 +

√−303)/2, GP/PARI’s [43] ‘bnfsunits’ gives as a basis (modulo torsion) the

set of elements{−20 + 3ω, 2,−4− ω,−36− ω, 4− ω, 28− ω,−12 + ω}

of norms 210, 22, 25 · 3, 27 · 11, 23 · 11, 25 · 13, and 24 · 13, respectively.Then ker(δ2,k) is a free Z-module of rank several hundreds, but most of the elements of a Z-

basis for this kernel turn out to result in the trivial element of p(k), i.e., correspond to relationsof the type (3.21) and (3.22). In this case, with a set of exceptional units that differed fromthe one used in Appendix B, but again using that |K2(O)|= 22, we found using the approachdescribed in §6.2 that the element

−46[−ω − 27

64] + 36[

−ω + 15

16] − 14[

−ω + 41

16] − 48[

−ω + 8

4] − 62[

−ω + 41

4] + 18[

−ω + 41

48] + 34[

−11

2]

+ 42[−143

1] − 16[

−253ω + 2321

6144] + 28[

−253ω − 495

3328] + 158[

−3ω + 23

32] + 4[

−39ω − 221

512] + 120[

−5ω + 73

64]

+ 18[−5ω + 49

416] + 44[

ω + 91

64] + 70[

ω + 27

64] − 12[

ω + 91

128] − 36[

ω + 1

16] + 82[

ω − 2

2] + 92[

ω + 7

32]

− 36[ω + 40

4] − 22[

ω − 8

4] − 116[

11

2] + 58[

11

8] − 34[

13

2] − 84[

13

4] + 34[

3ω + 20

8] + 14[

9ω − 17

352]

of Z[k[] belongs to ker(δ2,k), and that its image in B(k) is sent by ψk to a generator of K3(k)indtf .


Example 6.6. We now set k = Q(√−4547), so that O = Z[ω] with ω = 1+

√−45472 . We recall

that in [10] it was conjectured that |K2(O)| should be equal to 233. In fact, while the programdeveloped in loc. cit. showed that |K2(O)| divides 233, the authors were unable to verify theirconjecture since this would have required them to work in a cyclotomic extension of too high adegree.

By using the approach described in §6.2, we were now able to verify that |K2(O)| is indeedequal to 233 and, in addition, that the element

132[−2ω + 5

117] − 2[

−1

12] + 8[

−2ω − 3

1404] − 6[

−2ω + 5

18] − 14[

−2ω + 1752

19683] − 2[

−1

2] + 8[

−2ω − 3

27] + 2[

−1

288]

+ 2[−1

3] − 24[

−2ω + 1752

3159] − 2[

−1

36] + 128[

−2ω + 5

39] + 24[

−2ω − 3

4212] + 74[

−2ω + 5

52] + 12[

−2ω + 5

54]

− 12[−2ω − 840

6591] − 54[

−ω + 421

351] + 40[

−2ω − 3

72] − 16[

−2ω − 3

78] + 12[

−2ω + 5

8] − 10[

−ω + 421

468]

+ 2[−13

16] + 4[

−13

18] − 6[

−13

24] + 2[

−13

243] − 2[

−13

3] − 2[

−13

4] − 8[

−169

324] − 2[

−4ω − 1680

177957] − 6[

−208

81]

− 2[−26] − 4[−26

3] + 2[

−27

4] + 42[

−3ω + 1263

2197] − 12[

−31ω + 162

13182] + 22[

−31ω − 131

2197] − 4[

−5ω + 6

64]

− 8[−16ω + 6736

2197] + 2[

−16ω − 4523

2197] − 26[

2ω − 5

1404] + 2[

2ω + 3

18] + 2[

1

18] − 4[

ω + 875

1053] − 4[

ω + 875

1296]

− 24[2ω − 5

27] + 12[

2ω + 1750

3159] − 2[

1

32] + 14[

2ω − 5

351] − 14[

2ω − 5

4212] − 50[

2ω + 3

52] − 78[

2ω + 3

54]

+ 4[2ω − 842

6591] + 38[

ω + 420

351] − 30[

2ω − 5

72] − 2[

2ω − 5

78] − 14[

2ω + 3

8] + 14[

ω + 420

4056] − 6[

ω + 420

468]

− 6[117] − 2[13

256] − 2[

13

81] − 4[

169

16] − 14[

169

243] + 4[

169

256] − 10[

3494ω − 13298

2197] − 16[

4ω − 1684

177957]

− 42[16ω + 4523

8788] + 2[

243

256] + 2[

26

27] − 2[

26

9] + 4[

3

32] − 16[

3ω + 1260

2197] + 10[

31ω + 131

13182] − 24[

31ω − 162

2197]

− 6[39

2] + 6[

39

8] + 8[

8ω + 3360

351] + 20[

8ω + 3360

9477] − 30[

10ω + 2

1053] − 38[

5ω + 1

54] + 6[

5ω + 1

64] − 12[

5ω − 6

78]

− 4[5ω + 1

1053] + 24[

5ω + 1

27] + 8[

10ω − 12

729] − 6[52] − 4[

52

81] − 2[

64

81] − 14[

16ω + 4523

4563] + 4[

841ω − 176104

177957]

of Z[k[] belongs to ker(δ2,k), and that its image in B(k) is sent by ψk to a generator of K3(k)indtf .

Appendix A. Orders of finite subgroups

The main aim of this subsection is to prove, in Corollary A.5, that the lowest common multipleof the orders of finite subgroups of PGL2(O) divides 24. This result also implies the analogousstatement for stabiliser orders of polytopes arising from the quotient Γ\H3.

The authors are not aware of a suitable reference for this in the literature, or in fact, of anexplicit classification of types of finite subgroups of PGL2(O) that do not lie in PSL2(O). Asthis is not difficult, we include it for the sake of completeness.

Using the inclusions PSL2(O) ⊂ PGL2(O) ⊂ PGL2(C) = PSL2(C), our arguments are basedon the following classical result [19, Chap. 2, Th. 1.6] that goes back to Klein [28].

Proposition A.1. A finite subgroup of PSL2(C) is isomorphic to either a cyclic group of orderm ≥ 1, a dihedral group of order 2m with m ≥ 2, A4, S4 or A5. Further, all of these possibilitiesoccur.

It seems that the finite subgroups of PSL2(O) have been studied more than those of PGL2(O)even though it is harder to determine them.

We note that an element γ in SL2(O) of finite order has a characteristic polynomial of theform x2 + ax + 1 with a in [−2, 2] ∩ O as the two roots must be roots of unity, and conjugate.Hence a = ±2, ±1 or 0, from which it follows readily that the image γ in PSL2(O) of γ musthave order 1, 2 or 3.


In view of Proposition A.1, this limits the possibilities of a finite subgroup of PSL2(O) to thecyclic groups of orders 1, 2, or 3, the dihedral groups of order 4 or 6, and A4.

Cyclic groups of order 2 or 3 can be obtained already in the subgroup PSL2(Z), generated by(0 −11 0

)and

(0 −11 −1

)respectively.

The occurrence of the dihedral groups of order 4 and 6, and of A4, in PSL2(O) for k = Q(√−d)

with d a positive square-free integer, depends on the prime factorisation of d; see [32, Satz 6.8.].An element in PGL2(O) of odd order is contained in PSL2(O): if an odd power of its deter-

minant is a square in O∗, then so is its determinant.It follows that the only finite subgroups that can occur in PGL2(O) but are not necessarily

contained in PSL2(O) are the cyclic groups of even order, the dihedral groups, and S4.In order to obtain a complete answer, we first look at elements of finite order in PGL2(k).

Lemma A.2. Let γ be an element of finite order in GL2(k), and γ its image in PGL2(k). Writeord(γ) and ord(γ) for the respective orders of these elements.

(i) For k = Q(√−1), one has ord(γ) ∈ {1, 2, 3, 4, 6, 8, 12} and ord(γ) ∈ {1, 2, 3, 4}.

(ii) For k = Q(√−2), one has ord(γ) ∈ {1, 2, 3, 4, 6, 8} and ord(γ) ∈ {1, 2, 3, 4}.

(iii) For k = Q(√−3), one has ord(γ) ∈ {1, 2, 3, 4, 6, 12} and ord(γ) ∈ {1, 2, 3, 6}.

(iv) For all other k, one has ord(γ) ∈ {1, 2, 3, 4, 6} and ord(γ) ∈ {1, 2, 3}.Proof. Assume γ has order n, and let a(x) in k[x] be its minimal polynomial, so that a(x)divides xn − 1. The statement is clear if a(x) splits into linear factors, so we may assume a(x)is irreducible in k[x] and of degree 2. If a(x) is in Q[x] then it is an irreducible factor in Q[x]of xn − 1, necessarily the nth cyclotomic polynomial as the mth cyclotomic polynomial dividesxm − 1. So ϕ(n) = 2, and n = 3, 4 or 6. If a(x) is not in Q[x] then a(x)a(x) is irreduciblein Q[x], it must be the nth cyclotomic polynomial, and k must be a subfield of Q(ζn). Thenϕ(n) = 4, so n = 8 and k = Q(

√−1) or Q(

√−2), or n = 12 and k = Q(

√−1) or Q(

√−3). (Note

that n = 5 is excluded as Q(ζ5) contains no imaginary quadratic field.) The statement aboutthe order of γ follows by taking into account the factorisation of xn − 1 over k[x]. In general,the 2mth cyclotomic polynomial divides xm + 1. But for k = Q(

√−1) and n = 12, so m = 6,

we also have x6 + 1 = (x3 − i)(x3 + i) in k[x], and a(x) divides one of those factors. �

Remark A.3. In [36], in the cell stabiliser calculation for k = Q(√−3), the symbol A4 should

be a dihedral group of order 12 in PGL2(O), where the subgroup in PSL2(O) is dihedral oforder 6.

We can now determine the types of finite subgroups in PGL2(O) that do not lie in PSL2(O).

Proposition A.4. Let G be a finite subgroup of PGL2(O) that is not contained in PSL2(O).

(i) For k not equal to Q(√−m) with m = 1, 2 or 3, G is isomorphic to a cyclic group of

order 2, or a dihedral group of order 4 or 6. For k = Q(√−1) or Q(

√−2), the list has

to be extended with a cyclic group of order 4, a dihedral group of order 8, and S4. Fork = Q(

√−3) the list has to be extended with a cyclic group of order 6, and a dihedral

group of order 12.(ii) All the groups listed in claim (i) occur.

Proof. We already observed that an element of finite odd order in PGL2(O) is contained inPSL2(O), so the groups listed in claim (i) are those that are not ruled out by combining Propo-sition A.1 with Lemma A.2.


It remains to show that all such groups occur. Various examples may of course exist in theliterature, but for the sake of completeness we give some here.

In fact, for S4 we use the stabiliser of the single element in Σ∗3 in our calculations for Q(√−1)

respectively Q(√−2).

Cyclic examples. If u in O∗ is not a square, then ( u 00 1 ) is not in PSL2(O) and its order equals

the order of u. This gives the required subgroups except for those of order 2 for Q(√−1), and

of order 4 for Q(√−2). The former can be obtained by using

(0√−1

1 0

), which is not in PSL2(O)

and has order 2, and the latter by using(

0 11√−2

), which is not in PSL2(O) and has order 4.

Dihedral examples. If u in O∗ is not a square, and has order m = 2, 4 or 6, then ( u 00 1 ) and

( 0 11 0 ) generate a dihedral group of order 2m. With u = −1 this constructs a suitable copy of D4,

except for Q(√−1), but for this field we can use generators

(−1 00 1

)and

(0 1√−1 0

). Taking u a

generator of O∗ gives a copy of D8 for Q(√−1), and a copy of D12 for Q(

√−3). For k not equal

to Q(√−1), a suitable copy of D6 is generated by

(0 −11 −1

)and ( 0 1

1 0 ), whereas for Q(√−1) we can

use(

0 −11 −1

)and

(1

√−1

1+√−1 −1

). Finally, a copy of D8 for Q(

√−2) is generated by

(0 11√−2

)and(

0 −11 0

).

S4-examples. For Q(√−1), the orders of

(1√−1+1

0 −1

),(√−1 −1−1 0

), and

(1√−1

0 −√−1

)are 2, 3, and

4, respectively, and they generate a subgroup isomorphic to S4.

For Q(√−2), the orders of

(−2 −

√−2−1

−√−2+1 1

),(−√−2−1 −

√−2+1

1√−2

), and

(2

√−2+1√

−2−1 −2

)are

3, 3, and 2, respectively, and they also generate a subgroup isomorphic to S4. �

Corollary A.5. The lowest common multiple of the orders of the finite subgroups of PGL2(O)is 24 if k is either Q(

√−1) or Q(

√−2) and is 12 in all other cases.

Proof. This is true for the groups listed in Proposition A.4, and the possible finite subgroups ofPSL2(O) (discussed before Lemma A.2) have order dividing 12. �

Appendix B. A generator of K3(k)indtf for k = Q(

√−303)

For an imaginary quadratic field k = Q(√−d), with ring of integers O, Browkin [9] has

identified conditions under which the order |K2(O)| is divisible by either 2 or 3 (for example, heshows that |K2(O)| is divisible by 3 if d ≡ 3 mod 9).

Moreover, all of the coefficients in the linear combination βgeo that occurs in Theorem 4.7(i)are divisible by 2 if k is not equal to either Q(

√−1) of Q(

√−2).

In addition, if k is also not equal to Q(√−3), then whilst Remark 5.3 shows that at least one

of the coefficients in βgeo is not divisible by 3 one finds, in practice, that most of these coefficientsare divisible by 3.

For these reasons, it can be relatively easy to divide βgeo by either 2 or 3. However, no sucharguments work when considering division by primes larger than 3 and this always requiresconsiderably more work.

It follows that if one uses the approach of §6.1, then any attempt to obtain a solution β inp(k), to the equation |K2(O)|·β = βgeo, or equivalently (taking advantage of Remark 4.9) to the

equation 2|K2(O)|·β = 2βgeo, in order to obtain a generator of B(k) is likely to be much more


Figure B.1. Hexagonal cap

difficult when |K2(O)| is divisible by a prime larger than 3. This observation motivates us todiscuss in some detail the field Q(

√−303).

For the rest of this section we therefore set

k := Q(√−303),

so that O = Z[ω] with ω = (1 +√−303)/2.

We recall that it was first conjectured in [10], and then verified in [2], that this field is theimaginary quadratic field of largest square-free discriminant for which |K2(O)| is divisible by aprime larger than 3. More precisely, this order was first conjectured and later determined to beequal to 22.

We apply the technique described in §4. The quotient PGL2(O)\H has volume

vol(PGL2(O)\H) = −π · ζ ′k(−1) ≈ 140.1729768601914879815382141215 . . . .

The tessellation of H consists of 132 distinct PGL2(O)-orbits of 3-dimensional polytopes:

• 87 tetrahedra• 29 square pyramids• 13 triangular prisms• 1 octahedron• 2 hexagonal caps—a polytope with a hexagonal base, 4 triangular faces, and 3 quadri-

lateral faces as shown in Figure B.1.

The stabiliser ΓP in PGL2(O) of each of these polytopes P is trivial except for eight poly-topes P . It is cyclic of order 4 for 4 triangular prisms, and cyclic of order 6 for 1 triangularprism, the octahedron, and both hexagonal caps. By Theorem 4.7, the tessellation and stabiliserdata give rise to an explicit element βgeo, which we compute by using the ‘conjugation trick’ ofRemark 4.9 as 1

2(2βgeo) (see Remark 5.4 for why we are allowed to divide by 2). The latter can

be written as a sum of 188 terms ai[zi], where ai is in 2Z and zi in k[.By Theorem 4.7 we have ∑

aiD(zi) = 24π · vol(PGL2(O)\H),

where D is the Bloch-Wigner dilogarithm.Using the algebraic approach described in § 4.4, we can find an element in B(k) with image

under the Bloch-Wigner function bounded away from 0, and it turns out that it suffices to restrictthe search to exceptional S-units where S consists of the prime ideals above {2, 3, 11, 13, 19}.Here 3 ramifies in O, and the other primes are the first four primes that split in O. One of


the combinations βalg found with smallest positive dilogarithm value is a sum of 110 terms,βalg =

∑bj [zj ], where the bj all happen to be ±2.

By comparing∑bjD(zj) with

∑i aiD(zi) above, we expect that α = βgeo − 22 · βalg should

represent the zero element in B(k). We can prove this by writing α explicitly as a sum of theelements specified in (3.21) and (3.22). A linear algebra calculation in Magma [8] shows that αcan be written as an integral linear combination of 1648 specializations of the 5-term relation,plus a good number of 2-term relations. These relations are only valid up to torsion elementsin B(k) but are precisely valid in the torsionfree (by Corollary 3.29) group B(k). Hence we canconclude that βgeo = 22 · βalg, as required.

It follows from Corollary 4.11(i) that the resulting element γalg = ψk(βalg) is a generator of

K3(k)tf . Note that this also shows that the homomorphsm ψk:B(k) → K3(k)indtf is bijective (as

predicted by Conjecture 3.33).The elements βalg and βgeo are given below. The five-term combinations are given explicitly

at https://mathstats.uncg.edu/yasaki/data/ .

βalg = −2[−ω + 41

52] − 2[

−ω + 2

6] − 2[

−ω − 1

6] − 2[

−ω + 92

64] − 2[

−ω + 701

676] − 2[

−ω + 12

8] − 2[

−ω + 4

8] − 2[

−ω − 3

1] − 2[

−ω + 8

12]

− 2[−ω + 12

16] − 2[

−ω + 15

16] − 2[

−ω + 2

16] − 2[

−ω − 11

16] − 2[

−ω − 25

2] − 2[

−ω + 15

22] − 2[

−ω + 26

22] − 2[

−3ω + 46

66]

− 2[−ω − 14

22] − 2[

−ω + 26

24] − 2[

−ω + 2

26] − 2[

−ω + 8

32] − 2[

−ω + 25

32] − 2[

−ω + 28

32] − 2[

−ω − 4

32] − 2[

−ω + 389

352]

− 2[−ω − 36

352] − 2[

−ω + 8

4] − 2[

−ω − 7

4] − 2[

−ω + 41

44] − 2[

−ω + 8

48] − 2[

−15ω + 147

104] − 2[

−4ω + 21

13] − 2[

−4ω + 21

33]

− 2[−21ω + 172

352] − 2[

−21ω + 201

352] − 2[

−23ω + 211

256] − 2[

−23ω + 124

312] − 2[

−2457ω − 611

22528] − 2[

−253ω + 2321

6144]

− 2[−253ω − 495

3328] − 2[

−27ω + 535

676] − 2[

−27ω + 168

676] − 2[

−27ω + 535

832] − 2[

−29ω + 1332

1331] − 2[

−29ω + 149

484]

− 2[−3ω + 84

64] − 2[

−3ω + 45

88] − 2[

−3ω + 46

88] − 2[

−3ω − 9

11] − 2[

−3ω + 36

16] − 2[

−3ω + 45

22] − 2[

−3ω + 46

22]

− 2[−3ω − 20

22] − 2[

−3ω − 21

22] − 2[

−3ω − 17

256] − 2[

−3ω + 15

32] − 2[

−3ω + 24

44] − 2[

−39ω − 221

512] − 2[

−8ω + 31

39]

− 2[−51ω + 3807

3328] − 2[

−12ω + 63

143] − 2[

−9ω + 17

26] + 2[

ω + 11

8] + 2[

ω + 3

8] + 2[

ω − 4

8] + 2[

ω + 25

1] + 2[

ω + 14

11]

+ 2[ω + 7

12] + 2[

ω + 11

16] + 2[

ω + 14

16] + 2[

ω + 1

16] + 2[

ω + 4

16] + 2[

ω − 25

16] + 2[

ω + 14

22] + 2[

ω + 25

24] + 2[

ω − 2

24] + 2[

ω + 14

26]

+ 2[ω − 2

3] + 2[

ω + 4

32] + 2[

ω + 7

4] + 2[

ω − 4

4] + 2[

ω + 11

48] + 2[

ω + 7

48] + 2[

15ω + 132

104] + 2[

15ω + 204

176] + 2[

21ω − 3

169]

+ 2[21ω + 151

352] + 2[

21ω + 180

352] + 2[

23ω + 45

256] + 2[

27ω − 51

484] + 2[

29ω + 335

512] + 2[

29ω − 1

176] + 2[

3ω − 123

121] + 2[

3ω + 33

13]

+ 2[3ω + 33

16] + 2[

3ω + 43

22] + 2[

3ω + 3

26] + 2[

3ω + 12

32] + 2[

3ω + 17

32] + 2[

3ω + 20

32] + 2[

3ω + 20

44] + 2[

33ω + 2651

5408]

+ 2[8ω + 23

39] + 2[

6399ω + 16348

42592] + 2[

7ω − 67

64] + 2[

7ω − 1

66] + 2[

7ω + 181

312] + 2[

9ω + 60

52] + 2[

9ω + 8

26] + 2[

9ω + 8

44].



βgeo = 108[ω + 4

6] + 36[

ω + 3

5] + 108[

ω + 11

13] + 36[

ω + 1

5] + 64[

ω + 3

4] + 24[

ω + 14

26] + 12[

3ω − 3

35] + 36[

3ω

38] + 64[

ω

4] + 180[

ω + 3

8]

+12[ω − 4

22]+24[

ω + 36

32]+88[

ω + 5

10]+36[

21ω + 192

689]+24[

ω − 4

8]+136[

3ω + 17

32]+180[

ω + 3

11]+12[

5ω + 44

104]+30[

9ω + 45

106]

+12[ω − 6

2]+88[

5ω + 23

48]+20[

5ω − 25

3]+12[

9ω + 38

38]+12[

ω + 19

20]+48[

ω + 4

24]+24[

5ω − 25

128]+24[

5ω + 148

128]+60[

ω + 24

26]

+ 24[42ω + 665

1121] + 24[

6ω + 95

77] + 12[

2ω + 17

33] + 12[

ω − 1

3] + 48[

ω − 1

19] + 24[

7ω − 28

55] + 48[

35ω + 644

984] + 24[

4ω + 24

59]

+12[4ω − 28

7]+24[

7ω + 76

55]+48[

7ω − 28

40]+60[

ω + 3

7]+12[

ω − 5

7]+24[

5ω − 25

56]+24[

ω + 11

9]+24[

9ω + 99

208]+60[

4ω + 8

41]

+ 24[15ω − 4

26] + 36[

ω + 27

26] + 36[

ω + 27

32] + 12[

2ω + 10

53] + 12[

2ω + 41

45] + 12[

ω + 75

76] + 6[

ω + 5

5] + 6[

5ω

81] + 12[

25ω + 1123

1298]

+ 12[11ω + 66

118] + 60[

4ω + 29

41] + 24[

15ω + 15

26] + 28[

5ω − 28

1272] + 12[

25ω + 1175

1166] + 12[

ω + 5

3] + 72[

3ω + 6

41] + 24[

35ω + 440

1007]

+ 24[3ω + 20

104] + 24[

ω + 58

54] + 24[

9ω + 522

583] + 24[

3ω + 20

11] + 12[

ω + 37

39] + 12[

6ω + 19

247] + 12[

15ω + 980

902] + 12[

ω + 7

10]

+ 24[3ω − 6

13] + 24[

3ω + 57

76] + 24[

ω + 36

44] + 12[

ω + 76

78] + 12[

27ω + 27

130] + 16[

3ω − 21

20] + 28[

3ω + 18

59] + 48[

5ω + 67

82]

+ 28[15ω + 1370

1298] + 28[

ω + 6

10] + 72[

3ω + 32

41] + 12[

8ω + 57

209] + 12[

ω + 2

3] + 24[

ω + 2

7] + 12[

7ω + 64

78] + 12[

16ω + 133

53]

+36[3ω + 15

53]+24[

ω + 4

7]+48[

2ω + 2

39]+12[

ω − 1

2]+12[

18ω + 684

779]+12[

9ω − 72

308]+12[

ω + 7

7]+48[

ω + 1

4]+24[

175ω + 1600

4134]

+ 30[9ω + 52

106] + 6[

25ω + 867

792] + 24[

7ω + 64

50]24[

36ω + 1876

2173] + 24[

ω + 40

44] + 24[

3ω + 20

14] + 24[

ω + 3

44] + 24[

7ω + 137

123]

+ 6[27ω − 28

26] + 6[

ω + 79

82] + 12[

9ω + 55

118] + 12[

16ω + 155

779] + 28[

ω + 6

6] + 4[

3ω + 17

5] + 16[

30ω + 627

1007] + 24[

4ω − 24

9]

+12[84ω + 1480

2173]+24[

9ω − 36

28]+12[

27ω + 988

826]+12[

7ω + 69

76]+24[

7ω − 28

198]+12[

21ω + 151

130]+12[

ω + 84

88]+6[

81ω + 331

574]

+12[9ω + 113

104]+24[

4ω + 31

59]+16[

11ω + 40

106]+8[

165ω − 765

11236]+8[

11ω − 51

15]+36[

21ω + 476

689]+28[

3ω + 274

295]+28[

5ω + 30

59]

+ 12[9ω − 54

7] + 8[

5ω − 20

9] + 8[

9ω − 36

440] + 4[

ω − 4

15] + 4[

5ω + 95

152] + 4[

3ω + 35

50] + 4[

ω + 19

25] + 16[

10ω + 209

159] + 20[

ω − 5

160]

+ 28[ω + 5

265] − 12[

−3ω

35] − 12[

−ω − 3

22] − 12[

−5ω + 49

104] − 12[

−ω − 5

2] − 12[

−9ω + 47

38] − 12[

−ω + 20

20] − 12[

−2ω + 19

33]

− 12[−ω3

] − 12[−4ω − 24

7] − 12[

−ω − 4

7] − 12[

−2ω + 12

53] − 12[

−2ω + 43

45] − 6[

−ω + 6

5] − 6[

−5ω + 5

81] − 12[

−25ω + 1148

1298]

− 12[−11ω + 77

118] − 12[

−25ω + 1200

1166] − 12[

−ω + 6

3] − 12[

−ω + 38

39] − 12[

−6ω + 25

247] − 12[

−15ω + 995

902] − 12[

−ω + 8

10]

− 12[−ω + 77

78] − 12[

−27ω + 54

130] − 16[

−3ω − 18

20] − 12[

−8ω + 65

209] − 12[

−ω + 3

3] − 12[

−7ω + 71

78] − 12[

−16ω + 149

53]

− 12[−ω2

]− 12[−18ω + 702

779]− 12[

−9ω − 63

308]− 12[

−ω + 8

7]− 6[

−25ω + 892

792]− 6[

−27ω − 1

26]− 6[

−ω + 80

82]− 12[

−9ω + 64

118]

− 12[−16ω + 171

779]− 4[

−3ω + 20

5]− 12[

−84ω + 1564

2173]− 12[

−27ω + 1015

826]− 12[

−7ω + 76

76]− 12[

−21ω + 172

130]− 12[

−ω + 85

88]

− 6[−81ω + 412

574] − 12[

−9ω + 122

104] − 12[

−9ω − 45

7] − 4[

−ω − 3

15] − 4[

−5ω + 100

152] − 4[

−3ω + 38

50] − 4[

−ω + 20

25];

References

[1] A. Ash, D. Mumford, M. Rapoport, and Y.-S. Tai, Smooth compactifications of locally symmetric varieties,second ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010, With the col-laboration of Peter Scholze.

[2] K. Belabas and H. Gangl, Generators and relations for K2OF , K-Theory 31 (2004), no. 3, 195–231.[3] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift I

(Boston), Prog. in Math., vol. 86, Birkhauser, 1990, pp. 333–400.[4] S. J. Bloch, Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph

Series, vol. 11, American Mathematical Society, Providence, RI, 2000, Manuscript from 1978 (“Irvine notes”).Published as volume 11 of CRM Monographs Series by American Mathematical Society.

[5] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. ENS 4 (1974), 235–272.[6] , Cohomologie de SLn et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa

Cl. Sci. (4) 4 (1977), no. 4, 613–636, Errata in vol. 7, p. 373 (1980).


[7] , Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4) 8 (1981), no. 1, 1–33.

[8] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. SymbolicComput. 24 (1997), no. 3-4, 235–265.

[9] J. Browkin, The functor K2 for the ring of integers of a number field, Universal algebra and applications(Warsaw, 1978), Banach Center Publ., vol. 9, PWN, Warsaw, 1982, pp. 187–195.

[10] J. Browkin and H. Gangl, Tame and wild kernels of quadratic imaginary number fields, Math. Comp. 68(1999), no. 225, 291–305.

[11] J. I. Burgos Gil, The regulators of Beilinson and Borel, CRM Monograph Series, vol. 15, Amer. Math. Soc.,Providence, RI, 2002.

[12] D. Burns and M. Flach, On Galois structure invariants associated to Tate motives, Amer. J. Math. 120(1998), no. 6, 1343–1397.

[13] , Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570(electronic).

[14] D. Burns and C. Greither, On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math.153 (2003), no. 2, 303–359.

[15] D. Burns, R. de Jeu, and H. Gangl, On special elements in higher algebraic K-theory and the Lichtenbaum-Gross conjecture, Adv. Math. 230 (2012), no. 3, 1502–1529.

[16] R. de Jeu, Zagier’s conjecture and wedge complexes in algebraic K-theory, Compositio Mathematica 96(1995), 197–247.

[17] M. Dutour Sikiric, H. Gangl, P. E. Gunnells, J. Hanke, A. Schurmann, and D. Yasaki, On the cohomology oflinear groups over imaginary quadratic fields, J. Pure Appl. Algebra 220 (2016), no. 7, 2564–2589.

[18] W. Dwyer and E. Friedlander, Algebraic and etale K-theory, Trans. Amer. Math. Soc. 292 (1985), no. 1,247–280.

[19] J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space: Harmonic analysis andnumber theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[20] M. Flach, On the cyclotomic main conjecture at the prime 2, J. Reine Angew. Math. 661 (2011), 1–36.[21] J.-M. Fontaine and B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et

valeurs de fonctions L, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc.,Providence, RI, 1994, pp. 599–706.

[22] J.-M. Fontaine, Valeurs speciales des fonctions L des motifs, Asterisque (1992), no. 206, Exp. No. 751, 4,205–249, Seminaire Bourbaki, Vol. 1991/92.

[23] A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991), Proc. Sympos.Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–96.

[24] E. Hecke, Vorlesungen uber die Theorie der algebraischen Zahlen, Chelsea Publishing Co., Bronx, N.Y., 1970,Second edition of the 1923 original, with an index.

[25] A. Huber and G. Kings, Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet charac-ters, Duke Math. J. 119 (2003), no. 3, 393–464.

[26] A. Huber and J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math.3 (1998), 27–133 (electronic), Erratum same volume, pages 297–299.

[27] B. Kahn, The Quillen-Lichtenbaum conjecture at the prime 2, preprint, 1997; available from http://www.

math.uiuc.edu/K-theory/208.[28] F. Klein, Ueber binare Formen mit linearen Transformationen in sich selbst, Math. Ann. 9 (1875), no. 2,

183–208.[29] F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det”

and “Div”, Math. Scand. 39 (1976), no. 1, 19–55.[30] M. Koecher, Beitrage zu einer Reduktionstheorie in Positivitatsbereichen. I, Math. Ann. 141 (1960), 384–432.[31] M. Kolster, T. Nguyen Quang Do, and V. Fleckinger, Twisted S-units, p-adic class number formulas, and

the Lichtenbaum conjectures, Duke Math. J. 84 (1996), no. 3, 679–717.[32] N. Kramer, Imaginarquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die

nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen, preprint, 2015; available from https://hal.

archives-ouvertes.fr/hal-00720823/en/.

http://www.math.uiuc.edu/K-theory/208

http://www.math.uiuc.edu/K-theory/208

https://hal.archives-ouvertes.fr/hal-00720823/en/

https://hal.archives-ouvertes.fr/hal-00720823/en/


[33] M. Le Floc’h, A. Movahhedi, and T. Nguyen Quang Do, On capitulation cokernels in Iwasawa theory, Amer.J. Math. 127 (2005), no. 4, 851–877.

[34] M. Levine, The indecomposable K3 of fields, Ann. Sci. Ecole Norm. Sup. (4) 22 (1989), no. 2, 255–344.[35] S. Lichtenbaum, Values of zeta-functions, etale cohomology, and algebraic K-theory, Algebraic K-theory,

II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst.,Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 489–501. Lecture Notes in Math., Vol. 342.

[36] E. R. Mendoza, Cohomology of PGL2 over imaginary quadratic integers, Bonner Mathematische Schriften[Bonn Mathematical Publications], 128, Universitat Bonn, Mathematisches Institut, Bonn, 1979, Dissertation,Rheinische Friedrich-Wilhelms-Universitat, Bonn, 1979.

[37] D. Quillen, Finite generation of the groups Ki of rings of algebraic integers, Algebraic K-theory 1, LectureNotes in Mathematics, vol. 341, Springer Verlag, Berlin, 1973, pp. 179–198.

[38] J. Rognes and C. Weibel, Two-primary algebraic K-theory of rings of integers in number fields, J. Amer.Math. Soc. 13 (2000), no. 1, 1–54, Appendix A by Manfred Kolster.

[39] P. Schneider, Uber gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), no. 2, 181–205.[40] C. Soule, K-theorie des anneaux d‘entiers de corps de nombres et cohomologie etale, Inventiones Mathemat-

icae 55 (1979), 251–295.[41] V. Srinivas, Algebraic K-theory, second ed., Progress in Mathematics, vol. 90, Birkhauser Boston Inc., Boston,

MA, 1996.[42] A. A. Suslin, K3 of a field, and the Bloch group, Trudy Mat. Inst. Steklov. 183 (1990), 180–199, 229,

Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and theirapplications (Russian).

[43] The PARI Group, Univ. Bordeaux, PARI/GP version 2.11.0, 2018, available from http://pari.math.

u-bordeaux.fr/.[44] G. Voronoi, Nouvelles applications des parametres continus a la theorie des formes quadratiques. Premier

memoire. Sur quelques proprietes des formes quadratiques positives parfaites, J. Reine Angew. Math. 133(1908), 97–102.

[45] L. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83,Springer-Verlag, New York, 1997.

[46] C. Weibel, Algebraic K-theory of rings of integers in local and global fields, Handbook of K-theory. Vol. 1,2, Springer, Berlin, 2005, pp. 139–190.

[47] , The norm residue isomorphism theorem, Topology 2 (2009), 346–372.

[48] C. Weibel, Etale Chern classes at the prime 2, Algebraic K-theory and algebraic topology (Lake Louise,AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993,pp. 249–286.

[49] C. A. Weibel, The K-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society,Providence, RI, 2013, An introduction to algebraic K-theory.

[50] D. Yasaki, Hyperbolic tessellations associated to Bianchi groups, Algorithmic number theory, Lecture Notesin Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 385–396.

[51] D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, Arithmetic algebraicgeometry (Texel, 1989), Progr. Math., vol. 89, Birkhauser Boston, Boston, MA, 1991, pp. 391–430.

http://pari.math.u-bordeaux.fr/

http://pari.math.u-bordeaux.fr/


King’s College London, Dept. of Mathematics, London WC2R 2LS, United KingdomEmail address: [email protected]

URL: https://nms.kcl.ac.uk/david.burns/

Faculteit der Betawetenschappen, Afdeling Wiskunde, Vrije Universiteit Amsterdam, De Boele-laan 1081a, 1081 HV Amsterdam, The Netherlands

Email address: [email protected]

URL: http://www.few.vu.nl/~jeu/

Department of Mathematical Sciences, South Road, Durham DH1 3LE, United KingdomEmail address: [email protected]

URL: http://maths.dur.ac.uk/~dma0hg/

Laboratoire de mathematiques GAATI, Universite de la Polynesie francaise, BP 6570 — 98702Faaa, French Polynesia

Email address: [email protected]

URL: https://gaati.org/rahm/

Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greens-boro, NC 27412, USA

Email address: d [email protected]

URL: https://mathstats.uncg.edu/yasaki/

https://nms.kcl.ac.uk/david.burns/

http://www.few.vu.nl/~jeu/

http://maths.dur.ac.uk/~dma0hg/

https://gaati.org/rahm/

https://mathstats.uncg.edu/yasaki/

hyperbolic tessellations and k3 - king's college londonnumber conjecture that was formulated...

Documents